How do you determine how to solve y=10x-8,y=1/2x+5
since both equations are y = ....
set them equal to each other.
10x - 8 = (1/2)x + 5
everybody times 2
20x - 16 = x + 10
19x = 26
x = 26/19
back into one of them
y = 10x-8
= 260/19 - 152/19 = 108/19
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To solve the system of equations y=10x-8 and y=1/2x+5, you have two options: substitution or elimination.
1. Substitution Method:
- Start with either equation and solve for y in terms of x.
Equation 1: y = 10x - 8 (let's call it Eq1)
Equation 2: y = (1/2)x + 5 (let's call it Eq2)
- Substitute Eq1 into Eq2:
(10x - 8) = (1/2)x + 5
- Simplify and solve for x:
10x - (1/2)x = 5 + 8
20x - x = (2/2)x = 13
19x = 13
x = 13/19
- Substitute the value of x back into Eq1 to find y:
y = 10(13/19) - 8 = 130/19 - 8 = 130/19 - (8*19)/19 = (130-152)/19 = -22/19
Therefore, the solution to the system of equations is x = 13/19 and y = -22/19.
2. Elimination Method:
- Multiply both sides of Equation 1 by 2 to make the coefficients of x in both equations equal.
Equation 1: 2y = 20x - 16 (let's call it Eq1')
Equation 2: y = (1/2)x + 5
- Subtract Equation 2 from Equation 1' to eliminate y:
2y - y = 20x - 16 - (1/2)x - 5
y = 20x - (1/2)x - 16 - 5
y = (40/2)x - (1/2)x - 21
y = (39/2)x - 21
- Now you have a new equation: y = (39/2)x - 21
- Substitute this equation back into one of the original equations (Eq1) to solve for x:
(39/2)x - 21 = 10x - 8
- Simplify and solve for x:
(39/2)x - 10x = -8 + 21
(39/2 - 20/2)x = 13
(19/2)x = 13
x = 13 * (2/19) = 26/19
- Substitute the value of x back into Eq1 to find y:
y = 10(26/19) - 8 = 260/19 - 8 = 260/19 - (8*19)/19 = (260-152)/19 = 108/19
Therefore, the solution to the system of equations is x = 26/19 and y = 108/19.