Math CentralQuandaries & Queries


Question from Laura, a parent:

How do I change 3y=4x+1 into standard Ax+By=C standard form. This is my son's schoolwork. I homeschool him. I got the answer to the problem as 3y=4x+1, but it needs to be written in standard form and I know it needs to read 4x-3y=-1, but how do I get that answer into standard form. I need to know how to explain it to him. Can you show me? Thank you so much.

Hi Laura,

If you have an algebraic equation you can modify it to change the form of the equation and still preserve the fact that the two sides are equal. The two operations that produce such modifications are

  • add the same number to each side of the equation and

  • multiply each side of the equation by the same number.

You want to attain the form Ax + By = C, that is the x and y terms on the left side of the equation and a constant on the right side. You have 3y = 4x + 1 so you need to "move" the term 4x to the left side. This you can do by adding -4x to each side.

3y = 4x + 1, so

-4x + 3y = -4x + 4x + 1, which simplifies to

-4x + 3y = 1.

This is in standard form but it seems your son's textbook wants the coefficient of x to be positive not negative. This you can achieve by multiplying each side of the equation by -1.

-4x + 3y = 1, so

-1(-4x + 3y) = -1 × 1.

Multiplying -1 through the left side gives

(-1) × (-4) x + (-1) × (3) y = -1, or
4x - 3y = -1.

I hope this helps,

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