Variables on both sides

In 5x − 3 = 2x + 9, subtracting 2x from both sides leaves 3x − 3 = 9, an equation you already know how to solve.

Gather the variable terms on one side

Sometimes the variable shows up on both sides of the equals sign, as in 5x − 3 = 2x + 9. None of the earlier moves isolate x directly, because whatever you do, there is still an x on each side. The fix is to use the same properties of equality on a term with a variable in it. Subtract 2x from both sides. On the left, 5x − 2x = 3x; on the right, 2x − 2x = 0. The equation becomes 3x − 3 = 9, a two-step equation. Add 3 to get 3x = 12, divide by 3 to get x = 4.

Check in the original equation, because both sides contain x. Left: 5 × 4 − 3 = 17. Right: 2 × 4 + 9 = 17. They agree. Notice that the common value, 17, is not the solution; the solution is the x that produces it. On a graph, the left side y = 5x − 3 and the right side y = 2x + 9 are two lines with different slopes, and they meet at the point (4, 17). The x of that point is the answer, and the y is the value both sides share.

Which side should the variable go to?

You may collect the variable on either side, and the answer is the same. A practical habit is to move the smaller variable term, so the coefficient that remains is positive. In 3y + 8 = 7y − 4, subtract 3y from both sides to get 8 = 4y − 4. Add 4 to get 12 = 4y, and divide by 4 to get y = 3. If you subtract 7y instead, you get −4y + 8 = −4, then −4y = −12, then y = 3 again. Both roads are correct; the first just has fewer negative signs to watch.

Constants can be on both sides too. The order that works every time is: first collect the variable terms on one side, then collect the constants on the other side, then divide by the coefficient. Here is an everyday version. One phone plan costs 20 dollars a month plus 2 dollars per gigabyte, and another costs 32 dollars a month plus 1 dollar per gigabyte (both made-up prices). They cost the same when 2g + 20 = g + 32, which happens at g = 12 gigabytes. Below 12 the first plan is cheaper; above 12 the second one is.

The graph lab draws a smaller example, 3x − 2 = x + 4. Each side is a line, and the lines cross above x = 3, where both sides equal 7. Make a and c equal and the two lines become parallel. They never cross, so the equation has no solution. That case, and the case where the lines lie on top of each other, are the subject of the next unit.