Chapter 6.3, Problem 31P

How is the multiplication or division property of equality used in the elimination method? Are the properties always needed? Explain.

To determine

To define:The multiplication or division property of equality used in the elimination method.

Answer to Problem 31P

Multiplication or division property of equality are used in the elimination method and

these properties are not always needed if the coefficient of the variables is same.

Explanation of Solution

If we multiply or divide both sides of the equation by the same numbers, the side will always be equal.

When solving a mathematical system using elimination, we need some variables to have different coefficients that will cancel. Usually we need to multiply one or both numbers by a number to make this happen.

  $3x+4y=6$

  $2x-4y=4$

Here, $4y+\left(-4y\right)=0$ ,so we don’t have to multiply anything.

  $3x+4y=6$

  $x+y=4$

None of these have opposite coefficients. So, we can multiply the bottom equation by -3 to get the x to cancel out or -4 to get the y to cancel out. Thus, we take -3.

  $3x+4y=6$

  $-3x-3y=-12$

Now, $3x+\left(-3x\right)=0$

Therefore, multiplication or division property of equality is used in the elimination method.

These properties are not always needed if the coefficient is same.