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Systems of two linear equations in two variables: SAT Math notes
Solving two linear equations together, setting them up from word problems, and deciding when a system has one solution, no solution or infinitely many.
Study it with worked examples and 8 practice questionsPractise Systems of two linear equations in two variables questions
What this topic tests
A system is two equations that must both be true at the same time. Each linear equation is a line, and the solution of the system is the point where the two lines cross.
There are two main ways to solve by hand. Substitution works well when one equation already says y=… or x=…. Elimination works well when both equations look like ax+by=c: you add or subtract them so that one variable disappears. Desmos gives you a third way: graph both lines and click where they cross.
Two lines can cross once (one solution), be parallel (no solution), or be the very same line (infinitely many solutions). Many of the harder SAT questions ask which of these happens, or what a constant must be to make one of them happen.
Key ideas
- The solution (x,y) of a system is the point where the two lines meet.
- Different slopes: exactly one solution. Same slope but different intercepts: no solution. Same slope and same intercept: infinitely many solutions.
- In ax+by=c form, two equations are the same line if one is a multiple of the other, constant included. They are parallel if the x and y parts are multiples but the constant is not.
- If the question asks for x+y, 3x+3y or something similar, try adding or subtracting the two equations before solving fully.
- Word problems usually give a count equation (how many in all) and a value equation (price times number for each kind, added up, equals the total money).
The 3 kinds of Systems of two linear equations in two variables question
Sorted from every Systems of two 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. Solve the system
How to spot it. Two equations are given, or two lines are graphed, and you are asked for x, y, the point where they meet, or an expression such as x+y or xy.
How to answer it:
- If a graph is given, the solution is the point where the lines cross. Read its coordinates.
- If one equation is already solved for a variable, such as y=3x or x=2, substitute it into the other equation.
- Otherwise, use elimination. Multiply one or both equations so that one variable has matching coefficients, then add or subtract the equations to cancel it.
- Solve for the remaining variable, then plug it back in to find the other one.
- Answer exactly what is asked. If it is an expression like x+y, first check whether adding or subtracting the equations gives it directly.
- Desmos shortcut: type both equations and click the point where the lines cross.
The trap. Stopping after finding one variable. The question may want the other one, or x+y, and the value you found first is often one of the wrong choices. Also, when you subtract one equation from another, subtract every term, including the constants, and watch the signs.
2. One, none or infinitely many solutions
How to spot it. The question says the system has no solution or infinitely many solutions and asks for a constant, asks which equation could complete such a system, or asks how many times two lines intersect.
How to answer it:
- Put both equations in the same form: both as y=mx+b, or both as ax+by=c with the variables lined up.
- In y=mx+b form, compare. Different slopes: one solution. Same slope, different intercepts: no solution. Same slope and same intercept: infinitely many.
- In ax+by=c form, check whether one equation's x and y coefficients are the same multiple of the other's. If the constant follows that multiple too, it is the same line. If not, the lines are parallel.
- To find a constant, write down the match the question requires, such as equal slopes, and solve it.
The trap. For no solution, the slopes must match but the intercepts must not. An equation that is an exact multiple of the first, like x+3y=5 next to 2x+6y=10, gives infinitely many solutions, not none. Always check the constant as well as the slope.
3. Word problems with two unknowns
How to spot it. A story with two kinds of things, such as two ticket prices, two tent sizes or two products, and two separate totals. You either pick the system that models it or solve for one of the amounts.
How to answer it:
- Name the two unknowns, for example s for standard tickets and p for premium tickets.
- Write the count equation: the number of each kind adds up to the total number.
- Write the value equation: price (or size, or points) times number for each kind, added up, equals the total value.
- If you need a number, solve the system, usually by substituting from the count equation.
- Answer the question that was asked, such as the number of students rather than chaperones.
The trap. Putting the totals in the wrong equations, such as setting the number of tickets equal to the dollar total. Ask what each total measures: a count of things or an amount of money. Also make sure each price sits next to its own variable.
Practise Systems of two 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.
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