Question

"Use substitution to find the point that is a solution to the system of equations in each case. For the last two equations, isolate the y in the first equation first so that you know what to substitute into the second one. The systems are:

(2) y=8-x, 7=2-y
(3) y=-2x+6, 3y-x+3=0
(4) y-2x=3, 3x-2y=5
(5) 18=3x+y, 4x=3y-2

Please find the solution points for each of these systems using substitution."

Answer 1

By isolating y in each system of equations, substituting it into the second equation, and solving for x, we can find the corresponding y values to get the solution points for each system: (13, -5), (3, 0), (11, 25), and (6, 0).

Step-by-step explanation:

To solve each system of equations by substitution, we follow several steps. First, in the equations provided, we will solve for y when it is straightforward and substitute the expression for y into the other equation. Then, we solve for x and use the value of x to determine the corresponding value for y. Let's solve these systems one by one with the substitution method.

(2) System: y = 8 - x, 7 = 2 - y

Step 1: Isolate y in the first equation. This is already done.

Step 2: Substitute this expression for y into the second equation: 7 = 2 - (8 - x)

Step 3: Solve for x: 7 = 2 - 8 + x -> x = 7 - 2 + 8 -> x = 13

Step 4: Substitute this value of x back into the first equation to solve for y: y = 8 - 13 -> y = -5

The solution point is (13, -5).

(3) System: y = -2x + 6, 3y - x + 3 = 0

Step 1: Isolate y in the first equation. This is already done.

Step 2: Substitute this expression for y into the second equation: 3(-2x + 6) - x + 3 = 0

Step 3: Solve for x: -6x + 18 - x + 3 = 0 -> -7x + 21 = 0 -> x = 3

Step 4: Substitute this value of x back into the first equation to solve for y: y = -2*3 + 6 -> y = 0

The solution point is (3, 0).

(4) System: y - 2x = 3, 3x - 2y = 5

Step 1: Rearrange the first equation to isolate y: y = 2x + 3

Step 2: Substitute this expression for y into the second equation: 3x - 2(2x + 3) = 5

Step 3: Solve for x: 3x - 4x - 6 = 5 -> x = 11

Step 4: Substitute this value of x back into the first equation to solve for y: y = 2*11 + 3 -> y = 25

The solution point is (11, 25).

(5) System: 18 = 3x + y, 4x = 3y - 2

Step 1: Rearrange the first equation to isolate y: y = 18 - 3x

Step 2: Substitute this expression for y into the second equation: 4x = 3(18 - 3x) - 2

Step 3: Solve for x: 4x = 54 - 9x - 2 -> x = 6

Step 4: Substitute this value of x back into the first equation to solve for y: y = 18 - 3*6 -> y = 0

The solution point is (6, 0).