Question

Multiply each equation by a number that produces opposite coefficient for x or y 4x +5y =7

3x-2y=-12

Answer 1

To produce opposite coefficients for x and y in the given equations, we can multiply the first equation by -3 and the second equation by 4. This allows us to eliminate the x variable and solve for y. The solution to the system of equations is x = -2 and y = 3.

Step-by-step explanation:

To multiply each equation by a number that produces opposite coefficients for x or y, you need to find a number that, when multiplied by the coefficient, results in a value with the opposite sign. For example, if the coefficient is positive, you need to multiply by a negative number, and if the coefficient is negative, you need to multiply by a positive number.

In the given equations, we have:

4x + 5y = 7

3x - 2y = -12

To produce opposite coefficients for x, we can multiply the first equation by -3 and the second equation by 4. This gives us:

-12x - 15y = -21

12x - 8y = -48

Now, we can add these two new equations to eliminate the x variable:

-12x - 15y + 12x - 8y = -21 + (-48)

-23y = -69

To solve for y, we divide both sides of the equation by -23:

y = -69 / -23

y = 3

Now, we can substitute this value of y back into one of the original equations (let's use the first equation) to solve for x:

4x + 5(3) = 7

4x + 15 = 7

4x = 7 - 15

4x = -8

x = -8 / 4

x = -2

Therefore, the solution to the system of equations is x = -2 and y = 3.

Answer 2

To produce opposite x terms:
-3(4x + 5y = 7) ⇒ -12x -15y = -21
4(3x - 2y = -12) ⇒ 12x - 8y = -48
so if you add the equations the x terms will cancel out

To produce opposite y terms:
2(4x + 5y = 7) ⇒ 8x + 10y = 14
5(3x - 2y = -12) ⇒ 15x -10y = -60
so if you add the equations now the y terms will cancel out

What you are doing is finding the LCM (least common multiple) of the coefficients for the terms you want to cancel out...
The LCM for 4 and 3 (coefficients of x) is 12 so you multiple each by what is required to get -12 and +12.
The LCM for 5 and 2 (coefficients of y) is 10 so you multiple each by what is required to get -10 and +10.