Solve for the variable 1. 5x – 4 = 2x + 8 2. -4(x + 2) + 3x = 2.
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Transcript of Solve for the variable 1. 5x – 4 = 2x + 8 2. -4(x + 2) + 3x = 2.
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.