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Math  /  Algebra

QuestionSolve the system by the method of substitution. Check your solution(s) graphically. {x−3y=−9x+2y=1\left\{\begin{array}{l} x-3 y=-9 \\ x+2 y=1 \end{array}\right. (x,y)=(□)(x, y)=(\square)

Studdy Solution

STEP 1

1. We are given a system of two linear equations with two variables.
2. The method of substitution involves solving one equation for one variable and substituting that expression into the other equation.
3. The solution will be checked graphically to ensure accuracy.

STEP 2

1. Solve one of the equations for one variable.
2. Substitute the expression from Step 1 into the other equation.
3. Solve the resulting equation for the remaining variable.
4. Substitute back to find the other variable.
5. Check the solution graphically.

STEP 3

Choose the first equation to solve for x x :
x−3y=−9 x - 3y = -9
Solve for x x :
x=3y−9 x = 3y - 9

STEP 4

Substitute the expression for x x from Step 1 into the second equation:
(3y−9)+2y=1 (3y - 9) + 2y = 1

STEP 5

Simplify and solve for y y :
3y−9+2y=1 3y - 9 + 2y = 1
Combine like terms:
5y−9=1 5y - 9 = 1
Add 9 to both sides:
5y=10 5y = 10
Divide by 5:
y=2 y = 2

STEP 6

Substitute y=2 y = 2 back into the expression for x x :
x=3(2)−9 x = 3(2) - 9
Simplify:
x=6−9 x = 6 - 9
x=−3 x = -3

STEP 7

The solution to the system is:
(x,y)=(−3,2) (x, y) = (-3, 2)

STEP 8

Check the solution graphically by plotting both equations:
1. The first equation x−3y=−9 x - 3y = -9 can be rewritten as x=3y−9 x = 3y - 9 .
2. The second equation x+2y=1 x + 2y = 1 can be rewritten as x=−2y+1 x = -2y + 1 .
Plot both lines on a graph. The point of intersection should be at (−3,2) (-3, 2) .
The solution is:
(x,y)=(−3,2) (x, y) = (-3, 2)

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