Solve the system of two equations using elimination with multiplication.
4x−6y=−48
−2x+4y=30 (1 point)
Responses
(3, 9)
left parenthesis 3 comma 9 right parenthesis
(−21,−6)
left parenthesis negative 21 comma negative 6 right parenthesis
(−3, 6)
left parenthesis negative 3 comma 6 right parenthesis
(−33,−9)
(−3, 6)
To solve the system of equations using elimination with multiplication, we need to eliminate one variable by multiplying one or both of the equations so that the coefficients of the variable will become opposite in sign.
First, let's multiply the second equation by 2 to make the coefficients of x opposite:
-2x + 4y = 30
2 * (-2x + 4y) = 2 * 30
-4x + 8y = 60
Now we have the following system of equations:
4x - 6y = -48
-4x + 8y = 60
Adding these two equations will eliminate the x variable:
(4x - 6y) + (-4x + 8y) = (-48) + 60
-6y + 8y = 12
2y = 12
y = 6
Now substitute the value of y into one of the original equations, let's use the first equation:
4x - 6(6) = -48
4x - 36 = -48
4x = -48 + 36
4x = -12
x = -12/4
x = -3
Therefore, the solution to the system of equations is (x, y) = (-3, 6).
To solve the system of equations using elimination with multiplication, follow these steps:
Step 1: Multiply one or both equations by a constant to make the coefficients of either the x or y terms in the two equations the same, but with opposite signs. This will allow you to eliminate one variable when you add or subtract the equations.
In this case, let's eliminate the y variable. To do this, we can multiply the first equation by 2 and the second equation by 3:
Equation 1: 2(4x - 6y) = 2(-48) -> 8x - 12y = -96
Equation 2: 3(-2x + 4y) = 3(30) -> -6x + 12y = 90
Step 2: Add or subtract the two equations to eliminate the variable you chose. In this case, we'll subtract Equation 2 from Equation 1:
(8x - 12y) - (-6x + 12y) = -96 - 90
8x - 12y + 6x - 12y = -186
14x - 24y = -186
Step 3: Simplify the equation obtained in Step 2.
14x - 24y = -186 can be simplified by dividing each term by 2:
7x - 12y = -93
Now we have a new equation: 7x - 12y = -93.
Step 4: Solve the new equation for one variable. Let's solve for x by isolating it:
7x = 12y - 93
Divide both sides by 7:
x = (12y - 93)/7
Step 5: Substitute the expression found for x back into one of the original equations to solve for y. Let's substitute it into the first equation:
4(12y - 93)/7 - 6y = -48
Simplify this equation to solve for y. Once you find the value of y, substitute it back into the expression found for x to find the solutions.