Linear equations in one variable

On the test

Algebra is about 35% of the Math section, around 6 to 7 questions in each 22-question module. Math modules mix all four domains and go from easier to harder within the module, so Algebra shows up from the first question to the last.

13 to 15 of the 44 questions (about 6 or 7 in each module). All through the module, easier first and harder at the end.

How I approached it

A small story from Aime

When I look at a linear equation, I think about a market scale. A seller puts maize on one side and weights on the other, and the scale only works when both sides are equal. If she adds 2 kg on the left, she must add 2 kg on the right, or the scale tips. I do not guess. I make the same move on both sides until only the maize is left. A linear equation in one variable is the same scale. The letter is the unknown weight, and my job is to keep both sides balanced while I get the letter alone. On the SAT, I read the equation, do the same step to both sides, and check my answer by putting it back in. That habit wins these questions.

This is the most friendly skill on the whole test, because every step is something you already know from primary school. Be patient, keep both sides equal, and you will get these right.

What is inside this part

Solving for x

You are given one equation and must find the value of the unknown.

Two-step equations · Brackets and fractions · Letters on both sides

Word problems

A real-life story is turned into an equation with a fixed amount and a repeating amount.

Fees plus price per item · Total cost, distance or time

Number of solutions

You must decide, or choose a constant so that, the equation has one, none or infinitely many solutions.

No solution · Infinitely many solutions

Finding a constant

One solution is given, and you must find a missing number in the equation.

Substitute the solution · Solve for k or a

How to know a question is from this part

One equation with one letter, or a short story that gives a total.

The words "what is the value of x" or "what is the solution"
"Has no solution" or "infinitely many solutions"
A letter like k or a that is a constant in the equation
A story with a fixed fee and a price per item
Only one letter and no powers

The lesson

1. What is a linear equation in one variable?

An equation is a sentence with an equals sign. It says two things have the same value. For example, 2x + 3 = 11 says that 2x + 3 and 11 are equal.

A variable is a letter, like x, that stands for a number we do not know yet. Linear means the letter is only to the power 1. You will see x, but never x² or √x. One variable means there is only one different letter in the equation.

To solve the equation means to find the number that makes the sentence true. For 2x + 3 = 11, the answer is x = 4, because 2(4) + 3 = 11.

2. What the question is really asking

Most of the time the SAT asks one of three things. First: what is the value of x? Second: what is the value of an expression, like 3x + 1? Third: for what value of a constant (a number like k or a) does the equation have no solution, one solution, or infinitely many solutions?

Always read the last sentence first. It tells you what to find. Many students solve for x and then forget that the question wanted 2x − 3.

3. The balance rule

Whatever you do to one side, do to the other side. Add 5 to both sides. Divide both sides by 3. This keeps the equation true.

Our goal is to get the letter alone. We undo operations in the reverse order: first undo adding or subtracting, then undo multiplying or dividing.

Solve 3x + 7 = 22.

Show the steps

Undo the + 7: subtract 7 from both sides. 3x = 15.
Undo the × 3: divide both sides by 3. x = 5.
Check: 3(5) + 7 = 22. Correct.

x = 5

4. Brackets, fractions and letters on both sides

If there are brackets, multiply them out first. This is called distributing. 4(x + 3) becomes 4x + 12. Be careful with a minus sign: −2(x − 5) becomes −2x + 10, not −2x − 10.

If there are fractions, multiply every term on both sides by the bottom number. This clears the fractions.

If the letter is on both sides, move the letter terms to one side and the numbers to the other.

Solve (2x + 1)/3 = x − 4.

Show the steps

Multiply both sides by 3: 2x + 1 = 3(x − 4).
Distribute: 2x + 1 = 3x − 12.
Subtract 2x from both sides: 1 = x − 12.
Add 12 to both sides: 13 = x.

x = 13

5. Word problems

Many questions tell a story. Find the amount that does not change (a fixed fee) and the amount that repeats (a price per item). Then build the equation: fixed + (price × number) = total.

Then solve it exactly as above.

A taxi charges 1,500 RWF to start, plus 600 RWF for each kilometre. A trip cost 6,900 RWF. How many kilometres was the trip?

Show the steps

Write the equation: 1500 + 600k = 6900.
Subtract 1500: 600k = 5400.
Divide by 600: k = 9.

9 kilometres

6. How many solutions? (one, none, or infinitely many)

After you simplify, there are three possible endings. If you get x = a number, there is exactly one solution. If you get a false sentence like 5 = 7, there is no solution. If you get a true sentence like 4 = 4, there are infinitely many solutions.

When the SAT asks for the value of k that gives no solution, make the x terms equal on both sides. Then make sure the plain numbers are different. For infinitely many solutions, make both sides exactly the same.

For what value of k does k(x − 2) + 4x = 9x − 1 have no solution?

Show the steps