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Linear equations in one variable: SAT Math notes
Solving equations with one unknown, like 3x+5=20, and using them to model and answer real-world questions.
Study it with worked examples and 8 practice questionsPractise Linear equations in one variable questions
What this topic tests
A linear equation in one variable has a single unknown, and that unknown is never squared or under a root. Your job is almost always the same: undo what has been done to the variable, one step at a time, keeping both sides balanced, until the variable stands alone.
The SAT dresses this up in a few ways. Sometimes it asks for an expression like 3x+8 instead of x itself. Sometimes it hides the equation inside a story about costs, payments or growth. And sometimes it asks how many solutions an equation has, or what a constant must be for the equation to have no solution or infinitely many.
These are some of the most learnable points on the test. If you write each step neatly and check your answer by plugging it back in, you can get nearly all of them right.
Key ideas
- Whatever you do to one side of an equation, do to the other side too.
- Distribute first, then combine like terms on each side, then gather the variable terms on one side.
- Clear fractions by multiplying both sides by a common denominator. Clear decimals by multiplying by 10, 100 or 1,000.
- If the question asks for an expression such as 2x+6, look for a quick way to get it straight from the equation before solving for x.
- Once simplified, ax+b=cx+d has exactly one solution when a≠c, no solution when a=c but b≠d, and infinitely many solutions when a=c and b=d.
- In a word problem, the number multiplied by the variable is usually a rate (per hour, per item, each), and a number standing alone is usually a starting amount or a one-time fee.
The 4 kinds of Linear equations in one variable question
Sorted from every Linear equations in one variable question in the site's bank: each kind keeps coming up, and has a method that works every time.
1. Solve for the variable
How to spot it. The question gives one equation with one letter and asks for the value of that letter. There are often parentheses, fractions or decimals to deal with first.
How to answer it:
- Clear any fractions or decimals by multiplying every term on both sides by the same number.
- Distribute to remove parentheses.
- Combine like terms on each side separately.
- Move the variable terms to one side and the plain numbers to the other.
- Divide by the number in front of the variable, then plug your answer back in to check it.
- Desmos shortcut: type the equation exactly as given, using x as the letter. Desmos draws a vertical line at the solution, and you can click it to read the value. This is great for messy decimals.
The trap. Sign slips cause most mistakes. When you distribute a negative, such as -7(t+3), the negative must reach both terms: -7t-21, not -7t+21. Also be wary of a choice that matches a halfway step; the test likes to list the number you get just before the final division.
2. Find the value of an expression
How to spot it. The question asks for something like 3x+8, x-3 or 2(x-8) rather than x alone. Often that expression looks a lot like a piece of the given equation.
How to answer it:
- Compare the expression you want with the equation you have. Is it a multiple of one side, as 9x+6 is 3(3x+2)? Or does a chunk like (x+4) appear in the equation?
- If it is a multiple, multiply or divide the whole equation by that number.
- If a chunk repeats, treat the chunk as one unknown and solve for it directly.
- If you don't see a shortcut, solve for the variable as usual, then substitute it into the expression.
- Before you choose, reread the question to make sure you are giving the expression, not x.
The trap. The most common wrong answer is the value of x itself, because that is what you are used to finding, and the test almost always lists it as a choice. Underline what the question asks for and check your final number against it.
3. Word problems: write, solve or interpret
How to spot it. A short story about money, time, growth or counting. You may be asked to choose the equation that models it, to find a number, or to say what a number in a given equation means.
How to answer it:
- Name the unknown in words, for example, h is the number of hours.
- Sort the numbers. A starting amount or one-time fee stands alone. A rate (per hour, per item, each) gets multiplied by the unknown.
- Build the equation: starting amount plus or minus rate times unknown equals the total.
- If you need a number, solve the equation. If you need a meaning, point to that term in the equation and describe it with its units.
- Check that the answer makes sense, such as a whole number of people or a positive amount of time.
The trap. Watch for the 'after the first few' setup, such as one price for the first 2 hours and an hourly rate after that. The rate applies only to the extra hours, t-2, not to all t hours. In questions about meaning, don't mix up a rate (each, per) with a total.
4. How many solutions, or find the constant
How to spot it. The question asks how many solutions an equation has, or tells you the equation has no solution or infinitely many solutions and asks for a constant such as a, k or r.
How to answer it:
- Simplify both sides fully into the form ax+b=cx+d by distributing and combining like terms.
- Compare the x-coefficients a and c. If they are different, there is exactly one solution.
- If the x-coefficients match and the constants b and d match too, both sides are identical, so there are infinitely many solutions.
- If the x-coefficients match but the constants are different, the equation can never balance, so there is no solution.
- To find an unknown constant, set the coefficients equal (and, for infinitely many solutions, the constants too) and solve.
- Desmos check: graph y= the left side and y= the right side. One crossing point means one solution, parallel lines mean none, and a single overlapping line means infinitely many.
The trap. Skipping the simplifying. For example, 10(15x-9) and -15(6-10x) look different, but both expand to 150x-90. Also, no solution needs the constants to be different; if they turn out equal as well, the answer is infinitely many solutions instead.
Practise Linear equations in one variable
On One for All SAT every question has a step-by-step solution, worked both by hand and in Desmos, the calculator built into the digital SAT, and the notes have worked examples of each kind of question, then practice questions you can check as you go. It's free, and there's nothing to sign up for.
Open the Linear equations in one variable notes