Solve each of the following systems of equations, using either substitution or elimination.
x - 2y = 5 2x - 3y = 6
2x - 4y = 10 (multiplied by 2)
2x - 3y = 6
-y = 4 (subtracting bottom equation from top)
Solve for y then x.
To solve the system of equations using substitution, we'll solve one equation for one variable and substitute that expression into the other equation.
Step 1: Solve the first equation for x
x - 2y = 5
x = 5 + 2y
Step 2: Substitute x into the second equation
2(5 + 2y) - 3y = 6
10 + 4y - 3y = 6
10 + y = 6
y = 6 - 10
y = -4
Step 3: Substitute the value of y back into one of the original equations to find x
x - 2(-4) = 5
x + 8 = 5
x = 5 - 8
x = -3
So, the solution to the system of equations is x = -3 and y = -4.
Now let's solve the same system of equations using elimination.
Step 1: Multiply the first equation by 2 to match the coefficients of x
2(x - 2y) = 10
2x - 4y = 10
Step 2: Subtract the second equation from the first equation to eliminate x
(2x - 3y) - (2x - 4y) = 6 - 10
2x - 3y - 2x + 4y = -4
y = -4
Step 3: Substitute the value of y into one of the original equations to find x
x - 2(-4) = 5
x + 8 = 5
x = 5 - 8
x = -3
Again, the solution to the system of equations is x = -3 and y = -4.