How to solve a system of equations

What a system of equations is

One equation with two unknowns, such as x + y = 10, has endless solutions: 3 and 7, 4 and 6, 2.5 and 7.5, and so on. Add a second condition, such as x − y = 4, and only one pair fits both: x = 7 and y = 3. That pair is the solution of the system.

Every solution of a linear equation in x and y is a point on its line. A solution of the system has to lie on both lines at once, so it is the point where they meet. This picture explains why there are only three possible outcomes for two lines: they cross once, they never meet, or they are the same line.

You can always check a solution by substituting it into every original equation. If any one of them fails, it is not a solution.

Method 1: graphing

Write each equation in slope-intercept form, y = mx + b, graph both lines, and read the point where they cross. For y = x + 1 and y = −x + 5, the first line starts at 1 on the y-axis and climbs 1 per step; the second starts at 5 and drops 1 per step. They cross at (2, 3). Checking: 2 + 1 = 3 and −2 + 5 = 3.

Graphing shows what a solution means, and it is a good first look. Its weakness is precision: if the lines cross at (1.4, 2.7), a hand-drawn graph will not tell you that exactly. Use graphing to estimate, and algebra to get the exact answer.

Method 2: substitution

Substitution works best when one equation already has a variable on its own, such as y = 2x − 1, or when a variable has a coefficient of 1 and is easy to isolate. Replace that variable in the other equation with its expression, solve the resulting one-variable equation, then substitute back.

Method 3: elimination

Elimination, also called the addition method, works best when both equations are in the form ax + by = c. Multiply one or both equations so that one variable has opposite coefficients, such as 4y and −4y, then add the equations so that variable disappears. Multiplying a whole equation by a number does not change its solutions, for the same reason it keeps a single equation balanced.

No solution and infinitely many solutions

Sometimes both variables disappear at once. What is left tells you which special case you have:

• A false statement, such as 3 = −1, means no solution. The lines are parallel: same slope, different y-intercepts. For y = 2x + 3 and y = 2x − 1, setting them equal gives 2x + 3 = 2x − 1, then 3 = −1, which is never true. The system is called inconsistent.
• A true statement, such as 0 = 0, means infinitely many solutions. The two equations describe the same line. For 2x + 4y = 8 and x + 2y = 4, doubling the second gives the first exactly, so every point on x + 2y = 4 is a solution. The system is called dependent.
• A quick test before solving: write both equations in slope-intercept form. Different slopes mean exactly one solution. The same slope with different intercepts means none. The same slope and the same intercept means infinitely many.

Worked example: a word problem

Word problems with two unknowns and two facts are natural systems. Name each unknown, write one equation per fact, and say what the answer means at the end.

Common mistakes

Watch for these when solving systems:

• Stopping after one variable. The solution is a pair, so find both values.
• Multiplying only some terms of an equation, including forgetting the number on the right side.
• Dropping parentheses when substituting, so 8(50 − a) becomes 400 − a instead of 400 − 8a.
• Subtracting equations and losing a sign. If you are unsure, multiply by a negative number and add instead.
• Reading 0 = 0 as the answer x = 0. It means infinitely many solutions, not that a variable is zero.
• Checking in only one equation. The pair must work in both originals.

Choosing a method and checking your work

Choose substitution when a variable is already alone or has a coefficient of 1 or −1. Choose elimination when both equations are in standard form, especially when coefficients already match or are easy to match. Use graphing to see the situation, to estimate, or when a graphing tool is allowed.

Always check the final pair in both original equations, not in a rearranged version, which may already contain a mistake. For extra practice, solve the same system with two different methods and confirm that they agree. Systems with three equations and three unknowns, which come up in Algebra II, use the same elimination idea twice.