Warm-upSolve for the variable

1. 5x – 4 = 2x + 8 2. -4(x + 2) + 3x = 2

Solution: (-2, 5)

Graph to find the solution.

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

.

x = -22x + y = 1 ....y = -2x +

1

....

..Wait! It’s not in

slope-intercept form!

Is there an easier way?

Do you notice anything about

one of the equations?

x = -22x + y = 1x = -2

2(-2) + y = 1-4 + y = 1 +4 =+4 y = 5Do you know the value of x?

(-2

Do you know the value of y? , 5)

Solution: (-2, 5)

Graph to find the solution.

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

.

x = -22x + y = 1 ....

y = -2x + 1

....

..

Solve Systems of Equations by

What did we do to solve it that way?

Substitution

Because we have to

substitute to find the

solution.

Why do you think we call it the substitution

method?

Duh.

How do we know when substitution is the better

method to use when solving a system?

Why is substitution sometimes a better method to use when

solving a system?

1)One equation will be ISOLATED (it will have either x or y by itself) or can be solved for x or y easily.

2)SUBSTITUTE the expression from Step 1 into the other equation and solve for the other variable.

3)SUBSTITUTE the value from Step 2 into the equation from Step 1 and solve.

4)Your SOLUTION is the ordered pair formed by x & y.

5)CHECK the solution in each of the original equations.

Ex. 1

( 16)

x = -43x + 2y = 20

Why would substitution be a good method

to use?x = -4

3(-4) + 2y = 20-12 + 2y = 20

+12 =+12 2y = 32 __ __ 2 2 y = 16

x y

3x + 2y = 203x + 2(16) = 203x + 32 = 20

- 32 = -32 3x = -12 __ __

3 3 x = -4

-4,

How can we check to see if our solution is correct?

x = -4 (-4, 16)3x + 2y = 20

3x + 2y = 203(-4) + 2(16) = 203(-4) + 2(16) = 20

-12 + 32 = 20

20 = 20We are correct!

Ex. 2

(2 )

y = x – 1x + y = 3

Why would substitution be a good method

to use?y = x – 1x +x – 1= 3

2x – 1 = 3 + 1 = +1 2x = 4 __ __

2 2 x = 2

x y

y = x – 1y = 2 – 1y = 1

x + y = 32 + y = 3

y = 1

, 1

How can we check to see if our solution is correct?

y = x - 11 = 2 - 11 = 1

x + y = 32 + 1 = 3The values

work in both equations, so

we are correct!

y = x – 1 (2, 1)x + y = 3 x y

3 = 3

Ex. 3

(-2

3x + 2y = -12y = x - 1

y = x - 13x + 2(x – 1) = -123x + 2x – 2 = -12 5x – 2 = -12 + 2 = + 2 ___ ____ 5x = -10 5 5 x = -2 x y

y = x - 1y = -2 - 1y = -3

, -3)

Does it matter which equation we

use to substitute x?

Why would we want to use y = x – 1 instead of

3x + 2y = -12?

How can we check to see if our solution is correct?

3x + 2y = -123 (-2) + 2 (-3) = -12 -6 + -6 = -

12

y = x - 1-3 = -2 - 1

The values work in both equations, so

we are correct! -3 = -

3

3x + 2y = -12 (-2, -3)y = x – 1 x y

-12 = -12

Ex. 4

22)

x = 1/2y – 34x – y = 10

x = 1/2y – 34(1/2y – 3) – y = 10 2y – 12 – y = 10

y – 12 = 10 +12 = +12

y = 22

yx

Does it matter which equation we

use to substitute y?

x = ½(22) – 3x = 11– 3

x = 8

(8,

How can we check to see if our solution is correct?

x = ½ y - 38 = ½ (22) - 38 = 11 - 3

4x – y = 104(8) – 22 =

10

The values work in both equations, so

we are correct!

32 – 22 = 10

8 = 8

x = 1/2y – 3 (8, 22)4x – y = 10 x y

10 = 10

Ex. 5

No Solution

x = -5y + 43x + 15y = -1

x = -5y + 43(-5y + 4) + 15y = -1-15y + 12 + 15y = -1-15y + 12 + 15y = -1 12 = -1

Does 12 = -1?

What is our

answer?

How can we check to see if our solution is correct?

x = -5y + 4 No solution3x + 15y = -1

There aren’t any values to

substitute and check.

So, what do we do?

Ex. 5Anytime the answer is No Solution….

x = -5y + 43x + 15y = -1

x = -5y + 43(-5y + 4) + 15y = -1-15y + 12 + 15y = -1-15y + 12 + 15y = -1 12 = -1

Does 12 = -1?

Just go back and check

your work and make sure there is no solution.

Ex. 6

-1)

2x – 5y = 29x = -4y + 8

x = -4y + 82(-4y + 8) – 5y = 29 - 8y + 16 – 5y = 29 - 13y + 16 = 29 -16 = -16 -13y =

13 ___ ___ -13 -13 y = -1

x y

Does it matter which equation we

use to substitute y?

x = -4y + 8x = -4(-1) + 8x = 4 + 8x = 12

(12,

How can we check to see if our solution is correct?

2x – 5y = 292 (12) – 5 (-1) = 29 24 – (-5) = 29

x = -4y + 812 = -4 (-1) +

8The values

work in both equations, so

we are correct!

24 + 5 = 29

2x – 5y = 29 (12, -1)x = -4y + 8 x y

29 = 29

12 = 4 + 812 = 12

So……Why is substitution sometimes a better method to use when

solving a system?

How do we know when substitution is the better

method to use when solving a system?

1)One equation will be ISOLATED (it will have either x or y by itself) or can be solved for x or y easily.

2)SUBSTITUTE the expression from Step 1 into the other equation and solve for the other variable.

3)SUBSTITUTE the value from Step 2 into the equation from Step 1 and solve.

4)Your SOLUTION is the ordered pair formed by x & y.

5)CHECK the solution in each of the original equations.