solve for the variable 1. 5x – 4 = 2x + 8 2. -4(x + 2) + 3x = 2
DESCRIPTION
x = -2 2x + y = 1 What makes a system easy to solve by graphing? …when both equations are in slope-intercept form.TRANSCRIPT
![Page 1: Solve for the variable 1. 5x – 4 = 2x + 8 2. -4(x + 2) + 3x = 2](https://reader035.vdocuments.mx/reader035/viewer/2022062504/5a4d1b2c7f8b9ab0599993da/html5/thumbnails/1.jpg)
Warm-upSolve for the variable
1. 5x – 4 = 2x + 8 2. -4(x + 2) + 3x = 2
![Page 2: Solve for the variable 1. 5x – 4 = 2x + 8 2. -4(x + 2) + 3x = 2](https://reader035.vdocuments.mx/reader035/viewer/2022062504/5a4d1b2c7f8b9ab0599993da/html5/thumbnails/2.jpg)
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!
![Page 3: Solve for the variable 1. 5x – 4 = 2x + 8 2. -4(x + 2) + 3x = 2](https://reader035.vdocuments.mx/reader035/viewer/2022062504/5a4d1b2c7f8b9ab0599993da/html5/thumbnails/3.jpg)
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)
![Page 4: Solve for the variable 1. 5x – 4 = 2x + 8 2. -4(x + 2) + 3x = 2](https://reader035.vdocuments.mx/reader035/viewer/2022062504/5a4d1b2c7f8b9ab0599993da/html5/thumbnails/4.jpg)
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
....
..
![Page 5: Solve for the variable 1. 5x – 4 = 2x + 8 2. -4(x + 2) + 3x = 2](https://reader035.vdocuments.mx/reader035/viewer/2022062504/5a4d1b2c7f8b9ab0599993da/html5/thumbnails/5.jpg)
Solve Systems of Equations by
What did we do to solve it that way?
Substitution
![Page 6: Solve for the variable 1. 5x – 4 = 2x + 8 2. -4(x + 2) + 3x = 2](https://reader035.vdocuments.mx/reader035/viewer/2022062504/5a4d1b2c7f8b9ab0599993da/html5/thumbnails/6.jpg)
Because we have to
substitute to find the
solution.
Why do you think we call it the substitution
method?
Duh.
![Page 7: Solve for the variable 1. 5x – 4 = 2x + 8 2. -4(x + 2) + 3x = 2](https://reader035.vdocuments.mx/reader035/viewer/2022062504/5a4d1b2c7f8b9ab0599993da/html5/thumbnails/7.jpg)
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?
![Page 8: Solve for the variable 1. 5x – 4 = 2x + 8 2. -4(x + 2) + 3x = 2](https://reader035.vdocuments.mx/reader035/viewer/2022062504/5a4d1b2c7f8b9ab0599993da/html5/thumbnails/8.jpg)
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.
![Page 9: Solve for the variable 1. 5x – 4 = 2x + 8 2. -4(x + 2) + 3x = 2](https://reader035.vdocuments.mx/reader035/viewer/2022062504/5a4d1b2c7f8b9ab0599993da/html5/thumbnails/9.jpg)
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,
![Page 10: Solve for the variable 1. 5x – 4 = 2x + 8 2. -4(x + 2) + 3x = 2](https://reader035.vdocuments.mx/reader035/viewer/2022062504/5a4d1b2c7f8b9ab0599993da/html5/thumbnails/10.jpg)
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!
![Page 11: Solve for the variable 1. 5x – 4 = 2x + 8 2. -4(x + 2) + 3x = 2](https://reader035.vdocuments.mx/reader035/viewer/2022062504/5a4d1b2c7f8b9ab0599993da/html5/thumbnails/11.jpg)
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
![Page 12: Solve for the variable 1. 5x – 4 = 2x + 8 2. -4(x + 2) + 3x = 2](https://reader035.vdocuments.mx/reader035/viewer/2022062504/5a4d1b2c7f8b9ab0599993da/html5/thumbnails/12.jpg)
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
![Page 13: Solve for the variable 1. 5x – 4 = 2x + 8 2. -4(x + 2) + 3x = 2](https://reader035.vdocuments.mx/reader035/viewer/2022062504/5a4d1b2c7f8b9ab0599993da/html5/thumbnails/13.jpg)
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?
![Page 14: Solve for the variable 1. 5x – 4 = 2x + 8 2. -4(x + 2) + 3x = 2](https://reader035.vdocuments.mx/reader035/viewer/2022062504/5a4d1b2c7f8b9ab0599993da/html5/thumbnails/14.jpg)
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
![Page 15: Solve for the variable 1. 5x – 4 = 2x + 8 2. -4(x + 2) + 3x = 2](https://reader035.vdocuments.mx/reader035/viewer/2022062504/5a4d1b2c7f8b9ab0599993da/html5/thumbnails/15.jpg)
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,
![Page 16: Solve for the variable 1. 5x – 4 = 2x + 8 2. -4(x + 2) + 3x = 2](https://reader035.vdocuments.mx/reader035/viewer/2022062504/5a4d1b2c7f8b9ab0599993da/html5/thumbnails/16.jpg)
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
![Page 17: Solve for the variable 1. 5x – 4 = 2x + 8 2. -4(x + 2) + 3x = 2](https://reader035.vdocuments.mx/reader035/viewer/2022062504/5a4d1b2c7f8b9ab0599993da/html5/thumbnails/17.jpg)
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?
![Page 18: Solve for the variable 1. 5x – 4 = 2x + 8 2. -4(x + 2) + 3x = 2](https://reader035.vdocuments.mx/reader035/viewer/2022062504/5a4d1b2c7f8b9ab0599993da/html5/thumbnails/18.jpg)
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?
![Page 19: Solve for the variable 1. 5x – 4 = 2x + 8 2. -4(x + 2) + 3x = 2](https://reader035.vdocuments.mx/reader035/viewer/2022062504/5a4d1b2c7f8b9ab0599993da/html5/thumbnails/19.jpg)
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.
![Page 20: Solve for the variable 1. 5x – 4 = 2x + 8 2. -4(x + 2) + 3x = 2](https://reader035.vdocuments.mx/reader035/viewer/2022062504/5a4d1b2c7f8b9ab0599993da/html5/thumbnails/20.jpg)
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,
![Page 21: Solve for the variable 1. 5x – 4 = 2x + 8 2. -4(x + 2) + 3x = 2](https://reader035.vdocuments.mx/reader035/viewer/2022062504/5a4d1b2c7f8b9ab0599993da/html5/thumbnails/21.jpg)
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
![Page 22: Solve for the variable 1. 5x – 4 = 2x + 8 2. -4(x + 2) + 3x = 2](https://reader035.vdocuments.mx/reader035/viewer/2022062504/5a4d1b2c7f8b9ab0599993da/html5/thumbnails/22.jpg)
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?
![Page 23: Solve for the variable 1. 5x – 4 = 2x + 8 2. -4(x + 2) + 3x = 2](https://reader035.vdocuments.mx/reader035/viewer/2022062504/5a4d1b2c7f8b9ab0599993da/html5/thumbnails/23.jpg)
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.