One for All SAT › SAT notes › Algebra › Linear equations in two variables
Linear equations in two variables: SAT Math notes
Writing, reading and graphing equations like 4x+3y=24 or y=mx+b, whose solutions form a straight line.
Study it with worked examples and 8 practice questionsPractise Linear equations in two variables questions
What this topic tests
A linear equation in two variables, such as y=2x+3 or 4x+3y=24, has endless solutions. Each solution is a pair (x,y) that makes the equation true, and when you plot all of them you get a straight line.
The SAT tests this in two settings. In a story, an equation like 12.50g+1.50t=80 describes two quantities that share a total, and you write it, use it, or explain what its numbers mean. On a graph, you find a line's slope and intercepts, write its equation from points, a table or a picture, and work with parallel and perpendicular lines.
Most questions come down to two tools: slope-intercept form, y=mx+b, and the idea that a rate times an amount gives a total.
Key ideas
- Slope is rise over run: m=(y2-y1)/(x2-x1). In y=mx+b, m is the slope and (0,b) is the y-intercept.
- To read the slope of Ax+By=C, solve for y. The slope works out to -A/B.
- Find the x-intercept by setting y=0. Find the y-intercept by setting x=0.
- Parallel lines have equal slopes. Perpendicular slopes are negative reciprocals: flip the fraction and change the sign.
- In a story equation ax+by=c, the numbers a and b are usually a rate or price per item, ax and by are partial totals, and c is the grand total.
- Moving a line up or down changes only its y-intercept. To move y=mx+b down 5 units, subtract 5 from the right side.
- A point lies on a line exactly when its coordinates make the equation true.
The 4 kinds of Linear equations in two variables question
Sorted from every Linear equations in two variables question in the site's bank: each kind keeps coming up, and has a method that works every time.
1. Write or use an equation from a story
How to spot it. A story with two kinds of things that add up to a total: two ticket types, two mixtures, two activities. You either pick the equation that models it, or plug in one known amount and find the other.
How to answer it:
- Name the two unknowns and their units.
- For each kind, find its rate: price per item, points per token, percent of a mixture, calories per hour.
- Multiply each rate by its own unknown and add: first rate times x, plus second rate times y, equals the total.
- Check the units. Every term must measure the same thing as the total, such as dollars or pounds of salt.
- If one amount is given, substitute it and solve for the other.
The trap. Swapping the rates. If x is pounds of Fertilizer A, then A's rate must sit next to x. With percents, use the decimal (60% is 0.6), and make sure both sides measure the same thing: pounds of filler can't equal total pounds.
2. Interpret a number in the equation
How to spot it. The question gives an equation from a story and asks what a number, a term like 6y, or a point like (24,73) means. Some versions ask for a comparison, such as how much more one item is worth than another.
How to answer it:
- Write down what x and y stand for, with units.
- Identify the piece you are asked about. A number alone on one side is a total. A number multiplied by a variable is a rate per item. A product like 6y is a partial total.
- Describe it in words with units, for example, cups of jam in all the large containers.
- For a point (a,b), the first number is the value of x and the second is the value of y.
- Pick the choice that matches your description exactly.
The trap. Confusing a rate with a total. In 4x+6y=36, the 6 is cups per large container, but 6y is the cups in all the large containers together. Also keep the order of a point straight: (24,73) means x=24 and y=73.
3. Slope, intercepts and the equation of a line
How to spot it. Anything about a single line in the xy-plane: find its slope or an intercept, write its equation from a slope and a point, two points, a table or a graph, match an equation to a table, or move the line up or down.
How to answer it:
- If you are given two points or a table, find the slope: change in y divided by change in x, m=(y2-y1)/(x2-x1).
- Find b. Read the y-value where x=0, or plug one point into y=mx+b and solve.
- If you are given an equation in another form, such as Ax+By=C, solve it for y to read off m and b.
- For intercepts, set y=0 to find the x-intercept and x=0 to find the y-intercept.
- For a shift up or down by k units, add or subtract k on the right side of y=mx+b, then work with the new equation.
- Desmos shortcut: put the points in a table and type y1∼mx1+b to get the slope and intercept, or graph an equation and click where it crosses the axes.
The trap. Mixing up the intercepts. The y-intercept has x=0, so it is written (0,b); the x-intercept has y=0. Another trap is reading the slope of Ax+By=C as A or A/B. Solve for y and it comes out as -A/B.
4. Parallel and perpendicular lines
How to spot it. The words parallel or perpendicular appear, and you need a slope or the equation of a new line.
How to answer it:
- Find the slope of the given line. If it is written as Ax+By=C, solve for y first.
- For a parallel line, use the same slope.
- For a perpendicular line, flip the slope and change its sign. For example, -2 becomes 1/2.
- If the question wants an equation, use the new slope with the given point or intercept to write y=mx+b, then rearrange to match the choices.
- Special case: a vertical line like x=2 has no defined slope, so any line perpendicular to it is horizontal, with slope 0.
The trap. Doing only half the change for a perpendicular slope: flipping without changing the sign, or changing the sign without flipping. You need both. And get the original slope right first; for 4y+8x=6, the slope is -2, not 8 or 4.
Practise Linear equations in two variables
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 two variables notes