System of Equations: Definition, Solutions & Examples

Definitionnoun phrase

What is a system of equations in math?

A system of equations is a set of two or more equations that use the same variables and are solved together. Its solution is the values that make every equation true at the same time. For two linear equations, that is the point where the two lines cross. A system can have one solution, none, or infinitely many.

In Spanish: sistema de ecuaciones
Also called: simultaneous equations, system of linear equations, linear system
Grade level: 6-8

Grade 8Algebra 1CCSS 8.EE.C.8.aCCSS 8.EE.C.8.bCCSS HSA-REI.C.6GEDSATACT

System of Equations in a picture

The point (2, 3) is on both lines: 3 = 2 + 1 and 3 = −2 + 5. That is why it solves the system. Parallel lines never meet (no solution); identical lines meet everywhere (infinitely many).

Formal definition and notation

A system of k equations in the unknowns x₁, …, x_n is a list of equations that must hold at once. A solution is an assignment of values to all the unknowns that satisfies every equation. For two linear equations in x and y, the solutions correspond to the points of intersection of the two lines: one point, no point (parallel lines), or every point (the same line).

Symbol	Read it as	Meaning
{	brace	a large left brace often groups the equations of a system
(x, y)	the ordered pair x, y	a solution of a system in two variables
∅	empty set	the solution set when the system has no solution

Key facts

A point must satisfy every equation in the system. Fitting one equation is not enough.
There are three standard ways to solve a system: graphing, the substitution method and the elimination method. All three give the same answer.
A system of two linear equations never has exactly two solutions: two different lines meet at most once.
Write the solution of a two-variable system as an ordered pair (x, y), and check it in both original equations.
Word problems with two unknowns (adult and child tickets, two prices, two speeds) usually need two equations.

How many solutions? (two linear equations)

compare the slopes m and y-intercepts b of y = m₁ x + b₁ and y = m₂ x + b₂

m₁ ≠ m₂: the lines cross once: exactly one solution (consistent, independent)
m₁ = m₂ and b₁ ≠ b₂: parallel lines: no solution (inconsistent)
m₁ = m₂ and b₁ = b₂: the same line: infinitely many solutions (consistent, dependent)

When you solve algebraically, a false statement such as 3 = −1 means no solution, and a true statement such as 0 = 0 means infinitely many.

System of Equations examples, step by step

Example 1: Is the point a solution?

Warm-up

Is (3, −1) a solution of the system x + y = 2 and 2x − y = 7?

First equation: 3 + (−1) = 2. True.
Second equation: 2(3) − (−1) = 6 + 1 = 7. True.
The point makes both equations true.

Answer: Yes. (3, −1) is a solution because it satisfies both equations.

Example 2: Solve by substitution

Core

Solve the system y = 2x − 1 and x + y = 8.

The first equation already gives y. Replace y in the second: x + (2x − 1) = 8.
Combine like terms: 3x − 1 = 8, so 3x = 9 and x = 3.
Find y: y = 2(3) − 1 = 5.
Check in the second equation: 3 + 5 = 8. True.

Answer: The solution is (3, 5).

Example 3: Two unknowns in a word problem

Test-style

A school play sold 40 tickets for 375 dollars. Adult tickets cost 12 dollars and child tickets cost 7 dollars. How many adult tickets were sold?

Let a = adult tickets and c = child tickets. Count: a + c = 40.
Money: 12a + 7c = 375.
From the first equation, c = 40 − a. Substitute: 12a + 7(40 − a) = 375.
Simplify: 12a + 280 − 7a = 375, so 5a = 95 and a = 19.
Then c = 40 − 19 = 21. Check: 12(19) + 7(21) = 228 + 147 = 375.

Answer: 19 adult tickets were sold (and 21 child tickets).

Common mistakes

✗ Mistake(1, 5) solves y = x + 4 and y = 3x, because 5 = 1 + 4.

✓ InsteadCheck the second equation too: 3(1) = 3 ≠ 5. The solution is (2, 6).

A point must satisfy every equation in the system, not just one of them.

✗ MistakeSolving y = 2x − 1, x + y = 8 and answering x = 3.

✓ InsteadSubstitute back to get y = 5. The answer is the pair (3, 5).

A solution gives a value for every variable. One number is only half of it.

✗ Mistakey = 2x + 3 and y = 2x − 1 give 3 = −1, so the solution is 0 or x = −1.

✓ Instead3 = −1 is false for every x, so the system has no solution.

When the variables cancel and a false statement is left, the lines are parallel and never meet.

✗ Mistake4x − 2y = 6 and y = 2x − 3 give 6 = 6, so the solution is 6.

✓ Instead6 = 6 is always true, so there are infinitely many solutions: every point on y = 2x − 3.

A true statement with no variable left means the two equations describe the same line.

Not to be confused with

System of Equationsvs.Equation

One equation in two variables, such as x + y = 8, has infinitely many solutions: every point on its line. A system adds a second equation, which usually narrows the answer to one point.

System of Equationsvs.Substitution Method

A system is the problem: the equations themselves. Substitution is one method for solving it; graphing and elimination are others.

Quick check: 3 questions

Question 1 of 3

Which point is the solution of the system y = x + 4 and y = 3x?

Show answer and explanation

Answer: D. Set the right sides equal: x + 4 = 3x, so x = 2 and y = 6. Check: 6 = 2 + 4 and 6 = 3(2). Choice A fits only the first equation, B only the second, and C reverses the coordinates.

Question 2 of 3

How many solutions does the system y = 2x + 3 and y = 2x − 1 have?

Show answer and explanation

Answer: B. Both lines have slope 2 but different y-intercepts, so they are parallel and never meet. Algebra gives 2x + 3 = 2x − 1, or 3 = −1, which is false. Two lines can never meet exactly twice.

Question 3 of 3

What can you say about the system 4x − 2y = 6 and y = 2x − 3?

Show answer and explanation

Answer: C. Substitute: 4x − 2(2x − 3) = 6 becomes 6 = 6, always true. Both equations describe the same line, so every point on it works. Choices A and D are just two of those points (the intercepts).

System of Equations: frequently asked questions

What are the ways to solve a system of equations?

There are three common methods. Graphing: draw both lines and read the crossing point (good for estimates). Substitution: solve one equation for a variable and put that expression into the other. Elimination: add or subtract the equations to cancel a variable. All three give the same answer; pick the one that makes the arithmetic easiest.

How can you tell a system has no solution?

On a graph, the lines are parallel. In slope-intercept form, they have the same slope but different y-intercepts, as in y = 2x + 3 and y = 2x − 1. When you solve, the variables cancel and you get a false statement such as 3 = −1. Write the answer as no solution.

Can a system have exactly two solutions?

Not if both equations are linear: two different lines cross at most once. Systems with a curve can. For example, the line y = x and the parabola y = x² meet at (0, 0) and (1, 1), so that system has exactly two solutions.

Practice and tools

Systems of equations word problemsPractice on Effortless Math System of Equations CalculatorSolve a 2 by 2 system with steps Function GrapherGraph both lines and see where they cross Cramer's Rule CalculatorAnother way to solve linear systems

Sources

Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 8.EE.C.8.a, CCSS 8.EE.C.8.b, CCSS HSA-REI.C.6

Cite this entry

Effortless Math Team. “System of Equations.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/system-of-equations/

Flashcard

TermSystem of Equationstwo equations, one answer

DefinitionEquations solved together; the solution makes all of them true. y = x + 1 and y = −x + 5 meet at (2, 3).

Study the full deck

Browse all terms in the Math Dictionary · More algebra terms · Random term

EquationAn equation is a statement that two expressions are equal. It always has an equal sign, such as x + 5 = 12 or 3a = 18. Solving an equation means finding the value of the variable that makes it true: here x = 7 and a = 6. An expression like 4x + 7 has no equal sign, so it is not an equation.Algebra · Grade 6, Grade 7, Grade 8VariableA variable is a letter or symbol that represents a number or value, usually one that can change or is not yet known. In n + 5 = 12, the variable n is an unknown that equals 7. In A = lw, the variables l and w can take many values. Any letter can be a variable, not just x.Algebra · Grade 6, Grade 7, Grade 8SolutionA solution is a value that makes an equation or inequality true. To test a value, substitute it for the variable and check that the statement holds: x = 4 solves x + 2 = 6 because 4 + 2 = 6. A solution can also be an ordered pair, a point on a graph that satisfies the relationship.Algebra · Grade 6, Grade 7, Grade 8Linear EquationA linear equation is an equation whose graph is a straight line. In one variable, it usually has a variable to the first power, like 3x + 2 = 14. In two variables, it can be written in forms like y = mx + b, where m is the slope. No variable is squared, multiplied by another variable or in a denominator.Algebra · Grade 7, Grade 8, Algebra 1Parallel LinesParallel lines are lines in the same plane that never intersect. They stay the same distance apart and point in the same direction, like the rails of a straight railroad track. On a coordinate plane, parallel lines have the same slope but different y-intercepts. The symbol ∥ means is parallel to.Geometry · Grade 4, Grade 5, Grade 6Substitution MethodThe substitution method solves a system of equations by replacing one variable with an equivalent expression. Solve one equation for a variable, put that expression into the other equation, and solve the new one-variable equation. Then substitute back to find the other variable. It is easiest when one equation is already solved for a variable.Algebra · Grade 8, Algebra 1Elimination MethodThe elimination method solves a system of equations by adding or subtracting them so one variable cancels. Scaling one equation first is often needed to make the coefficients match. After one variable is eliminated, solve for the other variable, then substitute back to find the first.Algebra · Grade 8, Algebra 1Ordered PairAn ordered pair is two numbers written to locate a point on the coordinate plane. It is written as (x, y): the x-value comes first, and the y-value comes second. From the origin, move x units across, then y units up or down. The order matters: (2, 5) and (5, 2) are different points.Geometry · Grade 5, Grade 6, Grade 7Like TermsLike terms have exactly the same variables raised to exactly the same powers; only their coefficients may differ. 3x and 8x are like terms, and so are 5a² and −2a², but 4x and 4y are not. Only like terms can be combined by adding or subtracting their coefficients: 3x + 8x = 11x.Algebra · Grade 6, Grade 7, Grade 8SubstituteSubstitute means to replace a variable with a number or with an equal expression. In 4x − 1, substituting 3 for x gives 4(3) − 1. You substitute to evaluate expressions, to check whether a number is a solution, and to solve systems of equations by swapping in an equal expression.Algebra · Grade 6, Grade 7, Grade 8SimplifySimplify means to rewrite a number or expression in a shorter, easier form without changing its value. To simplify a fraction, divide the numerator and denominator by a common factor until none is left: 1824 = 34. To simplify an expression, combine like terms and do the operations you can: 3x + 2x = 5x.Number System · Grade 4, Grade 5, Grade 6NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2Slope-Intercept FormSlope-intercept form is the linear equation form y = mx + b, where m is the slope and b is the y-intercept. You can read both values directly from the equation without rearranging: in y = 2x + 5, the slope is 2 and the line crosses the y-axis at (0, 5).Algebra · Grade 8, Algebra 1SlopeSlope is a number that measures how steep a line is: the change in y divided by the change in x between any two points on the line, often called rise over run. A positive slope rises from left to right, a negative slope falls, and a horizontal line has slope 0.Algebra · Grade 8, Algebra 1Y-InterceptThe y-intercept is the point where a graph crosses the y-axis. Every point on the y-axis has x = 0, so you find the y-intercept by putting x = 0 into the equation. In y = mx + b, the y-intercept is b, written as the point (0, b).Algebra · Grade 8, Algebra 1ParabolaA parabola is the U-shaped curve that is the graph of a quadratic function such as y = x². It is symmetric about a line called the axis of symmetry, and its turning point is the vertex. Geometrically, it is every point equally far from a fixed point (the focus) and a fixed line (the directrix).Functions · Algebra 1, Algebra 2, Geometry

System of Equations

What is a system of equations in math?

System of Equations in a picture

Formal definition and notation

Key facts

How many solutions? (two linear equations)

System of Equations examples, step by step

Example 1: Is the point a solution?

Example 2: Solve by substitution

Example 3: Two unknowns in a word problem

Common mistakes

Not to be confused with

Quick check: 3 questions

System of Equations: frequently asked questions

What are the ways to solve a system of equations?

How can you tell a system has no solution?

Can a system have exactly two solutions?

Related terms

Practice and tools

Sources

Flashcard

Login

Login and use all of our services.

Register

Register Fast!

Choose your next math review topic