Expert: Josh - 1/31/2010

Question
if you times one equations by 2 shouldn't you times the other by 2  and i always get stuck at the substituting part i don't know what to use out   of the equation

Answer
Hi Brittnay,

It's difficult to talk about this without a concrete example.
But given two equations, e.g., solve
x+2y=5 ...[#1]
2x+y=4 ...[#2]
for x and y.

One way of finding the answers involving multiplying equation [#1] by 2, as a FIRST STEP.
This is permissible because it does not change the logic (or alter the truth) behind equation [#1]. In fact, as long as we are consistent, we can perform addition, subtraction or multiplication on this equation, as long as we add, subtract, or multiply the same quantity on BOTH sides of the equation.

This, however, does NOT mean that we have to multiply the second equation [#2] by the same amount.

Carrying the first step, as suggested, multiplying both sides of [#1] by 2, we get
2x+4y=10 ...let's call this [#3].

Next, we may subtract [#2] from [#3]:
2x+4y -(2x+y) = 10-4  this simplifies to
3y = 6
y = 2.

Finally, substituting this value of y into any equation, say [#1], we get x+2*2=5, x=1.

We do not always need to multiply both sides of an equation by some number to get closer to finding the answer. We do whatever is more convenient. The purpose of adding (subtracting, or multiplying) an equation by a number is to facilitate elimination of an unknown variable.

e.g., consider x+3y=6 ...[#1] and x+y=4 ...[#2]. In this case, to find "x" by eliminating "y", we can subtract x from both sides of [#2] to get y=4-x. Now, we can substitute this directly into [#1] (writing 4-x in place of y). This gives
x+3*(4-x)=6
12-2x=6
12-6=2x
x=6/2
x=3

Since y=4-x, y=1.