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Question Solve the system of linear equations 2x+4y=262x + 4y = 26 and 3x+3y=63x + 3y = 6 using elimination.

Studdy Solution

STEP 1

Assumptions1. We have two linear equations in two variables, x and y. . The system of equations is consistent and has a unique solution.

STEP 2

To use the elimination method, we need to make the coefficients of one of the variables the same in both equations. Looking at the equations, we can see that the coefficients of y are already the same. So, we can proceed to the next step.

STEP 3

Subtract the second equation from the first one to eliminate y.
(2x+y)−(3x+3y)=26−6(2x +y) - (3x +3y) =26 -6

STEP 4

Perform the subtraction operation.
(2x−3x)+(4y−3y)=26−6(2x -3x) + (4y -3y) =26 -6

STEP 5

implify the equation.
−x+y=20-x + y =20

STEP 6

Multiply the entire equation by -1 to get x on its own on the left side of the equation.
x−y=−20x - y = -20

STEP 7

Now we have a new system of equationsx−y=−203x+3y=6\begin{array}{l} x - y = -20 \\ 3x +3y =6\end{array}

STEP 8

Add the two equations together to eliminate y.
(x−y)+(3x+3y)=−20+6(x - y) + (3x +3y) = -20 +6

STEP 9

Perform the addition operation.
(x+3x)+(−y+3y)=−20+6(x +3x) + (-y +3y) = -20 +6

STEP 10

implify the equation.
4x+2y=−144x +2y = -14

STEP 11

Divide the entire equation by to isolate x.
x+y=−7x + y = -7

STEP 12

Substitute 2x+y=−72x + y = -7 into the second equation of the system to solve for y.
(2x+y)=6(2x + y) =6

STEP 13

implify the equation.
6x+3y=66x +3y =6

STEP 14

Subtract 6x6x from both sides of the equation.
3y=6−6x3y =6 -6x

STEP 15

Divide the entire equation by3 to isolate y.
y=2−2xy =2 -2x

STEP 16

Substitute y=2−2xy =2 -2x into the first equation of the system to solve for x.
2x+4(2−2x)=262x +4(2 -2x) =26

STEP 17

implify the equation.
2x+−x=262x + -x =26

STEP 18

Combine like terms.
−6x+8=26-6x +8 =26

STEP 19

Subtract8 from both sides of the equation.
−6x=26−8-6x =26 -8

STEP 20

implify the equation.
−6x=18-6x =18

STEP 21

Divide the entire equation by -6 to solve for x.
x=18/−6x =18 / -6

STEP 22

implify the equation.
x=−x = -

STEP 23

Substitute x=−3x = -3 into the equation y=−xy = -x to solve for y.
y=−(−3)y = -(-3)

STEP 24

implify the equation.
y=+6y = +6

STEP 25

implify the equation.
y=8y =8The solution to the system of equations is x=−3x = -3 and y=8y =8.

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