# Into Math Grade 8 Module 11 Lesson 1 Answer Key Prove the Pythagorean Theorem

We included HMH Into Math Grade 8 Answer Key PDF Module 11 Lesson 1 Prove the Pythagorean Theorem to make students experts in learning maths.

## HMH Into Math Grade 8 Module 11 Lesson 1 Answer Key Prove the Pythagorean Theorem

I Can prove the Pythagorean Theorem, use the Pythagorean Theorem to find unknown side lengths of right triangles, and identify a Pythagorean triple.

The given squares form a right triangle. Find the side lengths and areas of each. What patterns do you notice about the squares and side lengths? Right triangle is 3 cms, 4cms and 5 cms,
Squares, 1. 3 cms, 3 cms, 3 cms,
area = 9 cms, 2. 4 cms, 4 cms, 4 cms,
area = 16 cms, 3. 5 cms, 5 cms,5 cms,
area = 25 cms, Side lengths in squares increased by 1 cm and so areas increased,

Explanation:
Given squares are of lengths and areas of 1. 3 cms, 3 cms, 3 cms,
area = 3 cms X 3 cms = 9 cms, 2. 4 cms, 4 cms, 4 cms,
area = 4 cms X 4 cms = 16 cms, 3. 5 cms, 5 cms,5 cms,
area = 5 cms X 5 cms, 25 cms and the given squares form a right triangle let a,b and c be the side lengths as we know c2 = a2 + b2, So we have sides as 3 cms and 4 cms we get c2 = 32 + 42 , c2 = 25,
now value of c is square root of 25 = 5 cms. Side lengths in sqaures increased by 1 cm
and so areas increased.

Turn and Talk What types of objects in the real-world are in the shape of a right triangle? Can you find some right triangles in the classroom?
Answer A right angle is a 90 degree angle, so there are so many examples in real life like
Bermuda Triangle, Traffic Signs, Pyramids, Truss Bridges, Sailing Boat, Roof, Staircase and ladder,
Buildings, Monuments, and Towers,The height of a pole or a mountain, Sandwiches or Pizza Slices etc, Yes, some right triangles in the classroom are in the corners of a room, book, cube, windows and
at several other places,

Explanation:
A right angle is an abstraction like most objects in mathematics,
Right angles can be found in some of the most common shapes that we see every day,
Squares, rectangles, and right triangles all have right triangles,
Objects in the real-world are in the shape of a right triangle are Bermuda Triangle, Traffic Signs, Pyramids, Truss Bridges, Sailing Boat, Roof, Staircase and ladder, Buildings, Monuments, and Towers,The height of a pole or a mountain, Sandwiches or Pizza Slices etc, Yes,
some right triangles in the classroom are in the corners of a room, book, cube, windows and
at several other places.

Build Understanding

The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. If a and b are the lengths of the legs and c is the length of the hypotenuse, then a2 + b2 = c2.

Question 1.
Pythagoras was a Greek philosopher and mathematician whose name is given to the Pythagorean Theorem. You can use what you know about similar triangles to prove the Pythagorean Theorem. A. Using △ABC, draw a line from Point C perpendicular to the hypotenuse. Label the point where this line intersects the hypotenuse as Point D. This breaks the length c into two parts and forms two smaller triangles. Label AD as length e, and label DB as length f. Repeat the labels in the two smaller triangles.  Explanation:
Using △ABC, drawn a line from Point C perpendicular to the hypotenuse. Labelled the point where this line intersects the hypotenuse as Point D. This breaks the length c into two parts and forms two smaller triangles.
Labelled AD as length e and label DB as length f. Repeated the labels in the two smaller triangles as shown above.

B. Because of Angle-Angle Similarity, △ABC is similar to △CBD and △ACD. What do you know about the corresponding sides of similar triangles?
Two triangles are said to be similar if their corresponding angles are congruent and
the corresponding sides are in proportion,

Explanation:
Matching sides of two or more polygons are called corresponding sides, and matching angles are called corresponding angles. If two figures are similar,
then the measures of the corresponding angles are equal and the ratios of the lengths of the
corresponding sides are proportional.C. Use similar triangles to compare corresponding hypotenuses and corresponding longer legs in △CBD and △ABC. Use similar triangles to compare corresponding hypotenuses and corresponding shorter legs in △ABC and △ACD.  Explanation:
Used similar triangles to compare corresponding hypotenuses and corresponding longer legs in △CBD and △ABC as a/c = f/a and Used similar triangles to compare corresponding hypotenuses and corresponding shorter legs in △ABC and △ACD as c/b = b/e.

D. Use the Multiplication Property of Equality to rewrite the equations.
cf = __________ and b2 = ____________
cf = a2 and b2 = ce,

Explanation:
Using the Multiplication Property of Equality to rewrite the equations as cf = a2 and b2 = ce.

E. Use addition to write a2 + b2 in terms of c, e, and f.
Then simplify to complete the proof.
a2 + b2 = cf + ce, a2 + b2 = c(f + e), a2 + b2 = c2,

Explanation:
Using addition to write a2 + b2 in terms of c, e, and f.
Then simplifying to complete the proof as a2 + b2 = cf + ce,
a2 + b2 = c(f + e) as f + e = c so a2 + b2 = c X c, a2 + b2 = c2.

Step It Out

Question 2.
Use the Pythagorean Theorem to find the length of the set of stairs. A. Use the equation a2 + b2 = c2.
Substitute the leg lengths into the equation and simplify.
a2 + b2 = c2 + = c2 + = c2 = c2
52 + 122 = c2, 25 + 144 = c2, 169 = c2,

Explanation:
Given sides as a = 5 feet, 12 feet, Using the Pythagorean Theorem to find the length of the set of stairs and Using the equation a2 + b2 = c2.
Substituting the leg lengths into the equation and simplifying as 52 + 122 = c2,
25 + 144 = c2, 169 = c2.

B. Take the square root of both sides to solve for c, the hypotenuse.  = c
The length of the set of stairs is _________ feet.
Square root 169 = c2, c = 13, The length of the set of stairs is 13 feet,

Explanation:
Taking the square root of both sides to solve for c, the hypotenuse,
Square root 169 = c2, c = 13, The length of the set of stairs is 13 feet.

Question 3.
Only certain right triangles have side lengths that are all integers. The set of integers 5, 12, and 13 is one example of a Pythagorean triple, a set of integers that can form a right triangle. To find another common triple, find the length of the hypotenuse in the triangle shown.

Connect to Vocabulary
A Pythagorean triple is a set of positive integers a, b and c that fits the rule a2 + b2 = c2. a2 +b2 = c2 + = c2 + = c2 = c2 = c
The set of integers 3, 4, and ____________ is a common Pythagorean triple.
The set of integers 3, 4, and 5 is a common Pythagorean triple,

Explanation:
Given only certain right triangles have side lengths that are all integers. The set of integers 5, 12, and 13 is one example of a Pythagorean triple, a set of integers that can form a right triangle. To find another common triple, finding the length of the hypotenuse in the triangle shown as
32 + 42 = c2, 9 + 16 = c2, 25 = c2, c is equal to the sqaure root of 25 is 5,
therefore the set of integers 3, 4, and 5 is a common Pythagorean triple.

Turn and Talk Multiply 3, 4, and 5 each by 2. Is the new set of numbers a Pythagorean triple? Does this hold true for any multiple of 3, 4 and 5?
Yes, Yes, for all multiple of 3,4 and 5,

Explanation:
Multiplying 3, 4, and 5 each by 2. We get 6,8 and 10 the new set of numbers are not
a Pythagorean triple as 62 + 82 = c2, 36 + 64 = c2, 25 = c2,
Yes this hold true for any multiple of 3, 4 and 5, The set of Pythagorean Triples is endless,
We can prove this with the help of the first Pythagorean Triple (3, 4, 5):
Let n be any interger greater than 1, then 3n, 4n and 5n are also a set of Pythagorean Triple.
This is true because: (3n)2 + (4n)2 = (5n)
Examples:
If n it is  (3n,4n,5n)
If 2 it is (6,8,10), If 3 it is (9,12,15) and so on,
So we can make infinitely many triples just using the (3,4,5) triple
Question 4.
Use the Pythagorean Theorem to find the unknown leg length. Use the equation a2 +b2 = c2. Substitute the given lengths for one leg and the hypotenuse, and then simplify.
a2 + b2 = c2 + b2 =  + b2 = b2 = b = b = 15,

Explanation:
Using the equation a2 +b2 = c2.
Substituting the given lengths for one leg and
the hypotenuse and then simplifying as
a2 + b2 = c2,
82 + b2 = 172,
64 + b2 = 289,
b2 = 289 – 64 = 225, So b is equal to square root of 225,
we get b = 15.

Turn and Talk One leg of a right triangle is 9 inches and the hypotenuse is 21 inches. Find the length of the second leg. Do the lengths of the sides form a Pythagorean triple? Explain your reasoning.
The length of the second leg is 18.9
approximately 19,
Yes, the lengths of the sides form a
Pythagorean triple,

Explanation:
Given one leg of a right triangle is 9 inches and
the hypotenuse is 21 inches. Finding the length of
the second leg  let it be b as we know substituting as
a2 + b2 = c2,
92 + b2 = 212,
81 + b2 = 441,
b2 = 441 – 81 = 360,
So b is equal to square root of 360,
we get b = 18.97 approximately 19.
Yes the lengths of the sides form a
Pythagorean triple,
As we know Pythagorean triples (a,b,c) are three
non-negative integers that satisfy the condition of
Pythagoras theorem for a right-angled triangle.
As left side 92 + 18.972 = 81 + 359.86 = 440.86 ≈ 441
which is equal to right side therefore the lengths of
the sides form a Pythagorean triple.

Check Understanding

Question 1. Since the triangle is __________ and the ___________ lengths are 9 and 12 inches, by the Pythagorean Theorem, the hypotenuse length is ________ inches.
Since the triangle is right triangle and
the side lengths are 9 and 12 inches, by the Pythagorean
Theorem, the hypotenuse length is 15 inches,

Explanation:
Since the triangle is right triangle and
the side lengths are 9 and 12 inches by the Pythagorean
theorem  the hypotenuse length let it be c is
a2 + b2 = c2,
92 + 122 = c2,
81 + 144 = c2,
c2 = 225, So the value of c is square root of 225
which is 15 inches.

Question 2.
A. Find the unknown side length of the right-triangle sail. The unknown side length of the right-triangle sail is 5 feet,

Explanation:
Given the length for one side 12 feet and the hypotenuse 13 feet,
let b be unknown side length we have according to
Pythagorean theorem as a2 + b2 = c2, So substituting
122 + b2 = 132,
144 + b2 = 169,
b2 = 169 – 144 = 25 , So the value of b is square root of 25
which is 5 feet.

B. Do these lengths form a Pythagorean triple?
Yes,

Explanation:
As we know Pythagorean triples (a,b,c) are three
non-negative integers that satisfy the condition of
Pythagoras theorem for a right-angled triangle.
As left side we have 122 + 52 = 144 + 25 = 169
which is equal to right side 132 =169, therefore the
lengths of the sides form a Pythagorean triple.

Question 3.
Open-Ended In your own words, describe the Pythagorean Theorem and how you can use it.
Pythagorean theorem, the well-known geometric
theorem that the sum of the squares on the legs of
a right triangle is equal to the square on the hypotenuse,
Uses: Architecture and Construction,
Laying out square angles,
Surveying,
Solving two dimensional real life problems,

Explanation:
As we know Pythagorean theorem says the square of
the length of the hypotenuse of a right triangle
equals the sum of the squares of the lengths of the other two sides.
Uses of Pythagorean theorem are in
Architecture and Construction,
Laying out square angles,
Surveying,
Solving two dimensional real life problems,
Examples:
To calculate the length of staircase required to reach a window.
To find the length of the longest item can be kept in your room.
To find the steepness of the hills or mountains.
To find the original height of a tree broken due to
heavy rain and lying on itself.

Question 4.
Find the unknown length. The unknown length is 24 in,

Explanation:
Given the length for one side 7 in and the hypotenuse 25 in,
let b be unknown side length we have according to
Pythagorean theorem as a2 + b2 = c2, So substituting
72 + b2 = 252,
49 + b2 = 625,
b2 = 625 – 49 = 576 , So the value of b is square root of 576
which is 24 in.

Question 5.
A right triangle has leg lengths of 12 centimeters and 16 centimeters. What is the length of the hypotenuse?
The length of the hypotenuse is 20 centimeters,

Explanation:
Given a right triangle side lengthas leg lengths of
12 centimeters and 16 centimeters by the Pythagorean
theorem the hypotenuse length let it be c is
a2 + b2 = c2,
122 + 162 = c2,
144 + 256 = c2,
c2 = 400, So the value of c is square root of 400
which is 20 centimeters.

Question 6.
Critique Reasoning Rafael found the unknown side length of the given triangle using 5 the Pythagorean Theorem. Is he correct? Explain. 52 + 132 = c2
25 + 169 = c2
194 = c2
14 ≈ c
No, Rafael is incorrect,
The unknown side length is 12 not 14,

Explanation:
Given Rafael found the unknown side length of
the given triangle using  the Pythagorean Theorem.
Which is incorrect as instead of finding missing side he
substituted hypotenuse value as missing side the correct
method is let b be the missing side so
52 + b2 = 132,
25 + b2 =169,
b2 = 169 – 25 = 144, b equals to square root of 144 which is 12.

Question 7.
A right triangle has a leg length of 10 meters and a hypotenuse that is 26 meters. What is the length of the other leg?
The length of the other leg is 24 meters,

Explanation
Given a right triangle has a leg length of 10 meters and
a hypotenuse that is 26 meters. Let b is the length of
the other leg so 102 + b2 = 262,
100 + b2 = 676,
b2 = 676 – 100 = 576, b equals to square root of 576 which is 24.

Question 8.
Open-Ended What is a Pythagorean triple? Give one example.
A Pythagorean triple consists of three positive
integers a, b, and c, such that a2 + b2 = c2.
Such a triple is commonly written (a, b, c) and a
well-known example is (3, 4, 5),

Explanation:
A Pythagorean triple consists of three positive
integers a, b, and c, such that a2 + b2 = c2.
Such a triple is commonly written (a, b, c) and a
well-known example is (3, 4, 5) as
32 + 42 = 52,
9 + 16 = 25,
25 = 25 as both left side and right side are equal
therefore the set of integers 3, 4, and 5 is
a common Pythagorean triple.

Question 9.
Geography A land surveyor wants to measure the distance across a portion of a lake. The surveyor forms the right triangle shown by forming two legs of a right triangle measured on land. What is the distance across the lake? Show your work. The distance across the lake is 25 miles,

Explanation:
Given geography a land surveyor wants to
measure the distance across a portion of a lake.
The surveyor forms the right triangle shown by
forming two legs of a right triangle measured on land as
20 miles and 15 miles, Applying Pythagorean theorem
a2 + b2 = c2 , let the distance across the lake be c so
202 + 152 = c2,
400 + 225 = c2,
625 = c2,
So the value of c is square root of 625 which is 25 miles.

Question 10.
A sculpture shaped like a pair of right triangles has leg lengths of 8 feet and 15 feet. What is the length of the hypotenuse of the sculpture? The length of the hypotenuse of the sculpture is 17 feet,

Explanation:
Given a sculpture shaped like a pair of right triangles has
leg lengths of 8 feet and 15 feet. Let c be the length of
the hypotenuse of the sculpture applying Pythagorean theorem
a2 + b2 = c2 ,
82 + 152 = c2,
64 + 225 = c2,
289 = c2,
So the value of c is square root of 289 which is 17 feet.

Question 11.
Critique Reasoning Ashley claims she found the unknown side length of the triangle. Is she correct? Explain. 182 + 242 = c2
324 + 576 = c2
900 = c2
30 = c
No, Ashley is incorrect as the unknown side length is
15.87 ≈ 16 not 30,

Explanation:
Given one side length as 18 and hypotenuse as 24,
Ashley claims she found the unknown side length of
the triangle as 30 which is incorrect as the correct side
length is 182 + b2 = 242,
324 + b2 = 576,
b2 = 576 – 374 = 252, b equals to square root of 252
which is 15.87 approximately 16 not 30. So Ashley is incorrect.

I’m in a Learning Mindset!

What constructive feedback would you give Ashley to correct her misconception about the unknown side length of the triangle in Problem 11?
Would have substituted the value of hypotenuse value
correctly,

Explanation:
The constructive feedback would I give Ashley to
correct her misconception about the unknown
side length of the triangle in Problem 11 is that
she has substituted the value of hypotenuse in the
place of unknown side length and calculated again new
hypotenuse, if she has founded the unknown side length
substituting the given values correctly in the places,
She would have founded the value of unknown length correctly.

Lesson 11.1 More Practice/Homework

Question 1.
Open-Ended Draw a right triangle on graph paper. Measure the length of each leg and the hypotenuse. Write an equation using your triangle’s side lengths to show the Pythagorean Theorem holds true for your triangle.
Equation:
32 + 42 = 52,
Pythagorean Theorem holds true for my triangle,

Explanation:
Drawn a right triangle on graph paper. Measured
the length of each leg and the hypotenuse.
Wrote an equation using my triangle’s side lengths as
32 + 42 = 52, Now showing the Pythagorean Theorem
holds true for my triangle as we know the square of
the hypotenuse is equal to the sum of the squares of
the other two sides of a triangle.
As left side we have 32 + 42 = 9 + 16 = 25 which is equal to
right side 52 = 25. So my equation holds true.

Question 2.
A window shaped like a right triangle has the measurements shown. What is the length of the other leg? The length of the other leg is 12 feet,

Explanation:
Given a window shaped like a right triangle
has the measurements shown with one leg of
9 feet and hypotenuse as 15 feet, let b be the
unknown length of the other leg applying
Pythagorean theorem as 92 + b2 = 152,
81 + b2 = 225,
b2 = 225 – 81 = 144, b equals to square root of 144 which is 12 feet.

Question 3.
A right triangle has leg lengths of 7 inches and 24 inches. What is the length of the hypotenuse?
The length of the hypotenuse is 25 inches,

Explanation:
Given a right triangle has leg lengths of 7 inches and 24 inches.
Let c be the length of the hypotenuse applying
Pythagorean theorem as a2 + b2 = c2 ,
72 + 242 = c2,
49 + 576 = c2,
625 = c2,
So the value of c is square root of 625 which is 25 inches.

Question 4.
Math on the Spot Find each unknown side length.
A. The unknown side length hypotenuse is 10 cm,

Explanation:
Given a right triangle with side lengths as
8 cm and 6 cm, let the unknown hypotenuse side be c is
applying Pythagorean theorem
a2 + b2 = c2 ,
82 + 62 = c2,
64 + 36 = c2,
100 = c2,
So the value of c is square root of 100 which is 10 cm.

B. The unknown side length is 24 cm,

Explanation:
Given a right triangle with side lengths as
10 cm and hypotenuse side is 26 cm, Let b be the
unknown side length applying Pythagorean theorem
a2 + b2 = c2 ,
102 + b2 = 262,
100 + b2 = 676,
b2= 676 – 100 = 576,
So the value of b is square root of 576 which is 24 cm.

Question 5.
A. Find the unknown side length of a triangle with leg length 5 yards and hypotenuse length 13 yards.
The unknown side length is 12 yards,

Explanation:
Given to find the unknown side length of a triangle
with leg length 5 yards and hypotenuse length 13 yards.
Let unknown side be b applying Pythagorean theorem
a2 + b2 = c2 ,
52 + b2 = 132,
25 + b2 = 169,
b2= 169 – 25 = 144,
So the value of b is square root of 144 which is 12 yards.

B. Find the unknown side length of a triangle with leg length 30 yards and hypotenuse length 78 yards.
The unknown side length of a triangle is 72 yards,

Explanation:
Given to find the unknown side length of a triangle
with leg length 30 yards and hypotenuse length 78 yards.
Let the unknown side be b applying Pythagorean theorem
a2 + b2 = c2 ,
302 + b2 = 782,
900 + b2 = 6,084,
b2= 6,084 – 900 = 5,184,
So the value of b is square root of 5,184 which is 72 yards.

C. Use Structure How are the lengths of the leg and hypotenuse in Part B related to the lengths of the leg and hypotenuse in Part A?
The lengths of the leg and hypotenuse in Part B is
6 multiply of the lengths of the leg and hypotenuse in Part A,

Explanation:
If we see the lengths of the leg and hypotenuse in Part B
related to the lengths of the leg and hypotenuse in Part A as
we divide each by their lengths and hypotenuse we get
part B ( 30, 72, 78) and part A (5, 12, 13) we get
30/5 = 6, 72/12 = 6, 78/13 = 6 therefore the lengths of
the leg and hypotenuse in Part B is 6 multiply of the
lengths of the leg and hypotenuse in Part A.

D. Use Structure How is the length of the unknown side in Part B related to the length of the unknown side in Part A?
The length of the unknown side in Part B is 6 multiple
of the length of the unknown side in Part A,

Explanation:
We have the length of the unknown side in Part B is 72 yards and
the length of the unknown side in Part A is 12 yards
as 72/12 = 6 or 12 X 6 = 72. So the length of the unknown side in Part B is 6 multiple of the length of the unknown side in Part A.

Test Prep

Question 6.
What is the unknown side length? ___________ centimeters
The unknown side length is 12 cm,

Explanation:
Given a right triangle with side lengths as 5 cm and hypotenuse side is 13 cm, Let b be the
unknown side length applying Pythagorean theorem a2 + b2 = c2 ,
52 + b2 = 132, 25 + b2 = 169, b2= 169 – 25 = 144, So the value of b is square root of
144 which is 12 cm.

Question 7.
Which are examples of Pythagorean triples? Select all that apply.
(A) 6, 8, 10
(B) 8, 10, 12
(C) 10, 16, 15
(D) 12, 16, 20
(E) 14, 18, 20
(A) 6, 8, 10,
(D) 12, 16, 20,

Explanation:
Given to find the examples of Pythagorean triples as we know a Pythagorean triple consists of three
positive integers a, b, and c, such that a2 + b2 = c2 , So checking bit (A) 6, 8, 10 we have
62 + 82 = c2, 36 + 64 = c2, 100 = c2 and c is sqaure root of 100 which is 10  matches,
Checking with bit (B) 8, 10, 12 we have 82 + 102 = c2, 64 + 100 = c2, 164 = c2 and c is sqaure root of 164 which is 12.8 which is not matching with 12, Checking with bit (C) 10, 16, 15 we have
102 + 162 = c2, 100 + 256 = c2,
356 = c2 and c is sqaure root of 356 which is 18.86 which
is not matching with 15,
Checking with bit (D) 12, 16, 20 we have
122 + 162 = c2,
144 + 256 = c2,
400 = c2 and c is sqaure root of 400 which is 20 matches,
Checking with bit (E) 14, 18, 20 we have
142 + 182 = c2,
196 + 324 = c2,
520 = c2 and c is sqaure root of 520 which is 22.8 which is not matching with 20, therefore the examples of Pythagorean triples are bits (A) 6, 8, 10 and (D) 12, 16, 20.

Question 8.
A right triangle has leg lengths of 12 inches and 9 inches. What is the length of the hypotenuse?
__________ inches
15 inches,

Explanation:
Given a right triangle has leg lengths of 12 inches and 9 inches.
Let c be the length of the hypotenuse applying Pythagorean theorem
a2 + b2 = c2, 122 + 92 = c2, 144 + 81 = c2, 225 = c2,
So the value of c is square root of 225 which is 15 inches.

Question 9.
A right triangle has leg lengths of 7 feet and 8 feet. What is the length of the hypotenuse? Round to the nearest tenth.
_________ feet
The length of the hypotenuse is approximately 11 feet,

Explanation:
Given a right triangle has leg lengths of 7 feet and 8 feet.
Let c be the length of the hypotenuse applying Pythagorean theorem a2 + b2 = c2 ,
72 + 82 = c2, 49 + 64 = c2, 113 = c2, So the value of c is square root of 113 which is 10.63 feet
round to the nearest tenth is 11 feet.

Spiral Review

Question 10.
Solve the system of equations by graphing. Check your solution. y = 1 – x
y = 5 – 2x Point of intersection is at (4,-3), So values of x is 4 and y is -3,

Explanation:
Solving the system of equations y = 1 – x, y = 5 – 2x, y = 1 – x, so y = mx + b means
y = -x + 1, here m is -1 and y – intercept b = 1 and if x = 0, y = 1, x = 1, y = 0,
x = 2, y = -1, if x = 3, y = -2,if x = 4, y = -3, if x = 5,
y = -4, (0,1),(1,0), (2, -1), (3,-2) ,(4,-3),(5,-4),
Now y = 5 – 2x, y = -2x + 5, here m = -2 and y – intercept b = 5 and
if x = 0, y = 1, if x = 1, y= 3, if x = 2, y = 1, if x = 3, y = -1,
if x = 4, y = -3, if x =5, y = -5, So (0,5), (1,3),(2,1),(3,-1),(4,-3),(5,-5),
now pointing on graph above as shown we get point of intersection at (4,-3) now checking in equations as y = 1 – x, substituing x as 4 in y = 1- 4 = -3 which is true, now y as -3 we get -3 = 1 – x, x= 1 + 3 = 4 which is true, now checking in y = 5 – 2x substituting x as 4 in
y = 5 – 2 X 4 = 5 – 8 = -3  which is true and now y as -3 we get -3 = 5 – 2x, 2x = 5 + 3, x = 8/2 = 4 which is true, therefore, values of x is 4 and y is -3.

Question 11.
Order the numbers from least to greatest.
$$\frac{10}{4}$$, -3, $$\sqrt {8}$$ , 2.7
Least to greatest are -3, $$\frac{10}{4}$$, 2.7, $$\sqrt {8}$$,
Given to find order the numbers from least to greatest as $$\frac{10}{4}$$ = 2.5 and $$\sqrt {8}$$ = 2.82, So -3 < $$\frac{10}{4}$$ < 2.7 < $$\sqrt {8}$$,
therefore the order of the numbers from least to greatest are -3, $$\frac{10}{4}$$, 2.7, $$\sqrt {8}$$.