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QuestionSolve the system of equations: 2x−3y=72x - 3y = 7 and 4x−5y=134x - 5y = 13.

Studdy Solution

STEP 1

Assumptions1. The system of equations is linear and consistent, meaning it has at least one solution. . The variables x and y are real numbers.

STEP 2

We will solve this system of equations using the method of substitution or elimination. In this case, elimination method seems easier because we can easily eliminate one of the variables by manipulating the equations.

STEP 3

To use the elimination method, we need to make the coefficients of one of the variables the same in both equations. In this case, we can multiply the first equation by2 and the second equation by1 to make the coefficients of x the same.
2(2x−3y)=2(7)2(2x -3y) =2(7)1(x−5y)=1(13)1(x -5y) =1(13)

STEP 4

Perform the multiplication on both sides of the equations.
4x−6y=144x -6y =144x−y=134x -y =13

STEP 5

Now, we subtract the second equation from the first equation to eliminate x.
(4x−y)−(4x−5y)=14−13(4x -y) - (4x -5y) =14 -13

STEP 6

Perform the subtraction operation.
−y=1-y =1

STEP 7

To solve for y, we multiply both sides of the equation by -1.
y=−1y = -1

STEP 8

Now that we have the value of y, we can substitute it into the first equation to solve for x.
2x−3(−1)=72x -3(-1) =7

STEP 9

implify the equation by performing the multiplication.
2x+3=72x +3 =7

STEP 10

To solve for x, we subtract3 from both sides of the equation.
2x=7−32x =7 -3

STEP 11

Perform the subtraction operation.
x=4x =4

STEP 12

Finally, we divide both sides of the equation by2 to solve for x.
x=4/2x =4 /2

STEP 13

Perform the division operation.
x=2x =2So the solution to the system of equations is x =2 and y = -.

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