Multi-Step Linear Equations

When an equation has a bracket, expand it first. Three lots of x minus four means three times x and three times negative four, so the left side becomes three x minus twelve, and the equation is three x minus twelve equals fifteen. Add twelve to both sides to get three x equals twenty-seven, then divide both sides by three: x equals nine. Because the whole left side is three times something, you could also divide both sides by three first, giving x minus four equals five, and again x is nine. Both routes give the same answer.

Picture the equation as a level balance. The left pan holds five x's and a three; the right pan holds two x's and an eighteen. Take two x's off both pans and it stays level: three x plus three equals eighteen. Take three off both pans: three x equals fifteen. Split each side into three equal groups and x equals five. Check in the original: five fives plus three is twenty-eight, and two fives plus eighteen is also twenty-eight. Take away the smaller x-term, so the x's you keep are positive.

With brackets on both sides, expand both before you move anything. Four lots of x minus one is four x minus four, and two lots of x plus five is two x plus ten. Two x is the smaller x-term, so subtract two x from both sides: two x minus four equals ten. Add four: two x equals fourteen. Divide by two: x equals seven. Check: four times six is twenty-four, and two times twelve is twenty-four.

A negative x-term is easiest to remove by adding. In fourteen minus three x equals two x minus six, add three x to both sides: fourteen equals five x minus six. Add six: twenty equals five x, so x is four. Now a minus sign in front of a bracket. Two x minus, in brackets, x minus three, means subtract everything in the bracket, so both signs inside change: two x minus x plus three. That is x plus three equals ten, so x is seven.

Always check your answer by putting it back into the original equation. Solve six minus two lots of x minus three equals x. Negative two times negative three is positive six, so the left side is six minus two x plus six, which is twelve minus two x. Add two x to both sides: twelve equals three x, so x is four. Check: six minus two times one is four, and the right side is four. If you had made the sign slip and written minus six, you would get x equals zero, and the check catches it: the left side is twelve and the right side is zero. Quiz and exercises now.