Linear equations in one variable

What the SAT tests here

A linear equation in one variable has a single unknown, usually x, and that unknown is never squared, multiplied by itself or placed under a root. Your job is either to find the value of x or to decide how many solutions the equation has.

Expect three kinds of questions:

• Solve it: find the value of x, or of an expression such as 3x + 1.
• Translate it: turn a short real world description into an equation and solve.
• Count the solutions: decide whether the equation has one solution, no solution or infinitely many solutions, often with an unknown constant such as k.

The one rule behind every step

An equation is a balance. Whatever you do to one side, you do to the other side. Every move below is that rule applied in a smart order.

1. Clear parentheses by distributing.
2. Clear fractions by multiplying both sides by the least common denominator.
3. Combine like terms on each side.
4. Collect x on one side and numbers on the other.
5. Divide by the coefficient of x.
6. Check by substituting back. On the SAT this takes ten seconds and catches most sign errors.

Worked example 1: parentheses on both sides

Solve 3(x − 4) + 2 = 2x + 7.

• Distribute: 3x − 12 + 2 = 2x + 7
• Combine: 3x − 10 = 2x + 7
• Collect x: 3x − 2x = 7 + 10
• Result: x = 17

Check: the left side is 3(13) + 2 = 41 and the right side is 2(17) + 7 = 41. Both match.

Worked example 2: fractions

Solve x/3 + x/4 = 14.

The least common denominator of 3 and 4 is 12. Multiply every term by 12:

• 4x + 3x = 168
• 7x = 168
• x = 24

Check: 24/3 + 24/4 = 8 + 6 = 14.

Clearing fractions first turns a messy problem into a clean one. Multiply every single term, including the side without fractions.

Worked example 3: solve for an expression, not for x

If 2x + 5 = 11, what is the value of 6x + 15?

You could solve for x and substitute, but look at the target: 6x + 15 is exactly 3 times 2x + 5. So the answer is 3 × 11 = 33.

The SAT often asks for an expression on purpose. Before solving, ask whether the target is a multiple of what you already know.

One solution, no solution, infinitely many

Simplify both sides until the equation looks like ax + b = cx + d. Then compare:

What you see	Number of solutions	Example
Different x coefficients (a ≠ c)	exactly one	5x + 1 = 2x + 7
Same x coefficient, different constants	no solution	4x + 3 = 4x − 1
Same x coefficient, same constant	infinitely many	2(x + 3) = 2x + 6

If the x terms cancel and you are left with a false statement such as 3 = −1, there is no solution. If you are left with a true statement such as 6 = 6, every x works.

Worked example 4: a constant that controls the answer

In the equation 4(ax + 3) = 12x − 5, a is a constant. If the equation has no solution, what is the value of a?

• Distribute: 4ax + 12 = 12x − 5
• No solution means the x coefficients match but the constants do not.
• Match the coefficients: 4a = 12, so a = 3.
• The constants are 12 and −5, which are different, so the equation really has no solution.

Worked example 5: from words to an equation

A plumber charges a 45 dollar visit fee plus 30 dollars per hour. A customer paid 195 dollars in total. How many hours did the plumber work?

Let h be the number of hours.

• Fixed part plus rate times hours equals total: 45 + 30h = 195
• 30h = 150
• h = 5

In word problems, find the fixed amount and the amount that changes per unit. The fixed amount is the constant, the changing amount is the coefficient.

Common traps

• Distributing to only the first term. 3(x − 4) is 3x − 12, not 3x − 4.
• Losing a negative sign. −(x − 5) is −x + 5. Distribute the minus to every term.
• Multiplying only some terms when clearing fractions.
• Stopping at x when the question asks for something else, such as 2x or x + 3. Read the last line of the question again before you choose.
• Mixing up no solution and infinitely many. Same coefficients with different constants means none; everything identical means all.

Calculator tip

The Digital SAT gives you a built-in graphing calculator for the whole math section. For a one-variable equation, graph each side as its own line, for example y = 3(x − 4) + 2 and y = 2x + 7, and read the x value where they cross. Parallel lines mean no solution, the same line means infinitely many. Use it to check your algebra, not to replace it: solving by hand is usually faster.

Quick summary

• Distribute, clear fractions, combine, collect, divide, check.
• When a question asks for an expression, look for a shortcut multiple.
• Same x coefficient decides the special cases: different constants give no solution, equal constants give infinitely many.
• In word problems, the fixed amount is the constant and the per unit amount is the coefficient.