Systems of two linear equations in two variables
Algebra is about 35% of SAT Math (13β15 of 44 questions). Systems of two linear equations appear in both modules, as multiple choice and as typed answers, very often as a word problem with two unknown prices or amounts.
A solution is the crossing point
Each equation in a system is a line. The solution is the point that lies on both lines, which is where the lines cross. In the picture, the lines y = 2x β 1 and y = βx + 5 cross at (2, 3). Check: 3 = 2(2) β 1 β and 3 = β2 + 5 β. So the solution is x = 2, y = 3. The SAT may ask for x, for y, for x + y, or for the whole pair, so read the last line of the question.
Choose the method
Substitution and elimination both give exact answers. Choose by the look of the equations. If a variable is already alone (y = 3x β 4) or has coefficient 1, substitute. If both equations are in standard form (ax + by = c), eliminate. Whatever you do, find both variables and check the pair in both equations.
| Method | Use it when | Idea |
|---|---|---|
| Substitution | one variable is alone, or has coefficient 1 | replace it in the other equation |
| Elimination | both equations look like ax + by = c | add or subtract to cancel a variable |
| Graph / Desmos | the numbers are ugly, or to check | read the crossing point |
Elimination step by step
Make the coefficients of one variable equal or opposite. If they are already opposite (+2y and β2y), add the equations. If they are equal (+3x and +3x), subtract. If not, multiply one or both equations by a number first. Multiply every term, including the number on the right. After one variable is gone, solve for the other, then put it back into either equation to find the first.
One, none or infinitely many solutions
Two lines can cross once, be parallel, or be the same line. Compare slopes and intercepts. Different slopes: exactly one solution. The same slope but different intercepts: no solution. The same slope and the same intercept (one equation is a multiple of the other): infinitely many solutions. The SAT often asks for a missing constant: put the equations in the same form and compare the coefficients.
| Compare the lines | Example | Solutions |
|---|---|---|
| different slopes | y = 2x + 1 and y = βx + 7 | exactly one |
| same slope, different intercept | y = 2x + 1 and y = 2x + 5 | none |
| same slope, same intercept | 2x + y = 4 and 4x + 2y = 8 | infinitely many |
From a story to two equations
Name two unknowns and say what they mean. Most stories give two facts: a count (how many items in total) and an amount (how much money, weight or length in total). Write one equation for each fact. For example, with a adult tickets and c child tickets: a + c = 120 is the count, and 12a + 7c = 1160 is the money. Then solve the system. Finally, read the question again, because it may ask for only one unknown.
- The first equation has y alone, so substitute 3x β 4 for y in the second.
- 2x + (3x β 4) = 11
- 5x β 4 = 11, so 5x = 15 and x = 3.
- y = 3(3) β 4 = 5.
- Check in the second equation: 2(3) + 5 = 11 β
- Eliminate y. Multiply the first equation by 4 and the second by 3.
- 8x + 12y = 52
- 15x β 12y = β6
- Add: 23x = 46, so x = 2.
- From the first equation: 2(2) + 3y = 13, so 3y = 9 and y = 3.
- x + y = 5. Check in the second: 5(2) β 4(3) = β2 β
- Multiply the first equation by 2 to match the x-terms: 4x + 6y = 24.
- The second equation is 4x + ky = 20.
- With k = 6 the left sides are the same but 24 β 20, so the lines are parallel and never meet.
- k = 6. (If the right side were 24 instead of 20, there would be infinitely many solutions.)
- Let a = adult tickets and c = child tickets.
- Count: a + c = 120. Money: 12a + 7c = 1160.
- From the first, c = 120 β a. Substitute: 12a + 7(120 β a) = 1160.
- 12a + 840 β 7a = 1160, so 5a = 320 and a = 64.
- Then c = 56. Check: 12(64) + 7(56) = 768 + 392 = 1160 β
- Stopping after one variableFind both variables, then re-read the question. If it asks for x + y or for y, give exactly that.
- Subtracting equations and changing only some signsWhen you subtract, every term of the second equation changes sign, including the number on the right. Or add after multiplying one equation by β1.
- Multiplying only one side of an equationWhen you multiply an equation by 3, multiply every term, including the constant on the right-hand side.
- Mixing up βno solutionβ and βinfinitely manyβSame slope, different constants: no solution (parallel). Same slope and the matching constant (one equation is a multiple of the other): infinitely many.
- Mixing up the unknowns in a storyWrite βa = adult tickets, c = child ticketsβ before you set up the equations, and check that each equation uses the same meaning.
- Not checkingPut your pair into both original equations. It takes ten seconds and catches arithmetic slips.
Type both equations into Desmos, in any form: for example 2x + 3y = 13 on one line and 5x β 4y = β2 on the next. Click the crossing point and Desmos shows its coordinates. If the lines look parallel and never cross, there is no solution; if you see only one line, the equations describe the same line. For a missing constant k, type k in the equation: Desmos offers a slider, and you can move it until the lines become parallel or the same line.
- Type 2x + 3y = 13
- Type 5x β 4y = β2
- Click the crossing point: it is (2, 3)
- Type 4x + ky = 20 and drag the slider k until the line becomes parallel to the other equationβs line
Use Desmos for ugly numbers, to check a hand solution, and for questions about the number of solutions. For tidy numbers, elimination by hand is usually quicker.
Open Desmos β