Systems of linear equations — Digital SAT Math practice
Solve two equations in two variables; one, none or infinitely many solutions.
Key terms
- System of equations: Two or more equations with shared variables, solved for values that satisfy all of them.
- Substitution method: Solve one equation for a variable, then replace it in the other equation.
- Elimination method: Add or subtract scaled equations to cancel one variable.
- Consistent system: A system with at least one solution: one point, or infinitely many if the lines coincide.
- Inconsistent system: A system with no solution because the lines are parallel.
Common traps
- Assuming you must find both x and y when the question only asks for their sum or difference.
- Forgetting to distribute negative signs when subtracting equations during elimination.
- Confusing parallel lines with perpendicular lines, which have slopes that multiply to -1.
- Writing one equation for two unrelated quantities instead of defining variables first.
Key ideas to know
Solving by Substitution
Solve the system using substitution: y = 2x + 3 and 3x + y = 18. What is x?
Substitute the first equation into the second: 3x + (2x + 3) = 18. Combine like terms to get 5x + 3 = 18. Subtract 3 and divide by 5 to find x = 3.
Watch out: Forgetting to distribute or substitute the entire expression, especially when parentheses are needed around negative terms.
When to Use Elimination
Which method is fastest for this system? 4x + 5y = 23 and 4x − 2y = 9
Elimination. The coefficients of x are identical (4), so subtract the second equation from the first to cancel x immediately. Then solve for y.
Watch out: Scaling only one equation when you need to scale both to match coefficients before adding or subtracting.
Misinterpreting a Contradiction
A student solves a system and gets 0 = 9, then claims there are infinitely many solutions. What is wrong with this conclusion?
The conclusion is wrong. A false statement like 0 = 9 means the lines are parallel with no intersection, so there are zero solutions. An identity like 0 = 0 indicates infinitely many solutions.
Watch out: Confusing a false numerical statement (no solution) with an algebraic identity (infinitely many solutions).
Identifying Infinite Solutions
Solving a system yields the statement 0 = 0. What does this indicate about the number of solutions?
It indicates infinitely many solutions. The two equations represent the exact same line, so every point on the line satisfies both equations.
Watch out: Assuming a system must have one unique pair of values for x and y instead of recognizing dependent equations overlap completely.
Answer what is asked
Solve x + y = 10 and x − y = 2, then find the value of 3x − y.
Adding the equations gives 2x = 12, so x = 6 and y = 4. Then 3x − y = 3(6) − 4 = 14.
Watch out: Stopping at x or y when the question asks for a combination such as x + y or 3x − y.
Sample questions
Question 1
Adult tickets cost $10 and child tickets cost $6. A group buys 20 tickets for a total of $164. How many child tickets did they buy?
- 9
- 11
- 6
- 14
Show answer and explanation
Answer: 9
Let a + c = 20 and 10a + 6c = 164. Substituting c = 20 - a: 10a + 6(20 - a) = 164, so 4a = 44, a = 11 and c = 9.
Question 2
If 2x + 3y = 12 and x - y = 1, what is the value of x?
- 2
- 3
- 4
- 5
Show answer and explanation
Answer: 3
From x = y + 1: 2(y + 1) + 3y = 12, so 5y + 2 = 12, y = 2 and x = 3.