Systems of Linear Equations

Suppose a concert sells adult tickets and child tickets, and you know two facts: 200 tickets were sold, and the box office took in . Neither fact alone tells you how many of each were sold. Together they pin the answer down exactly. That is the idea behind a system of linear equations: several conditions that must all be true at once. This lesson covers what a system is, what counts as a solution, and how to tell whether a system has one solution, none, or infinitely many before you do any solving.

What Is a System of Equations?

A system of equations is a group of equations that share the same variables and are meant to be solved together. When every equation is linear, it is a system of linear equations. In general form, a system of two equations in two variables looks like this:

The curly brace is a reminder that the equations belong together. Here is a concrete example:

You’ll also hear these called simultaneous equations, especially in British textbooks. The name is a good description: the equations must hold simultaneously, for the same and the same .

What Is the Solution to a System of Equations?

A solution of a system is an ordered pair that makes every equation true. Making one equation true is not enough. Plenty of pairs satisfy , such as , and , but only one of them also satisfies .

To check a pair, substitute it into each equation in turn:

Try as well. It passes the first check () but fails the second (, not ), so it is not a solution. When a question says “if is the solution of the system of equations, find…”, it means the single pair that passes every check.

What a Solution Looks Like on a Graph

Every linear equation in two variables graphs as a line, and every point on that line is a pair that satisfies the equation. A solution has to lie on both lines, so it is the point where they cross:

That picture explains everything else about how many solutions a system can have.

How Many Solutions Can a System Have?

Two lines in a plane can relate to each other in only three ways, so a linear system has only three possible outcomes:

A linear system can never have exactly two or three solutions. If two different lines share two points, they are the same line, and then they share every point. The no-solution case is the one covered in more depth in the parallel lines lesson.

How to Tell the Number of Solutions Without Solving

Method 1: Compare Slopes and Intercepts

Rewrite each equation in slope-intercept form, , and compare:

For example, and have the same slope but different intercepts, so they are parallel and the system has no solution.

Method 2: Compare Coefficient Ratios

When both equations are in standard form, and , compare the ratios of matching coefficients:

Take with . All three ratios are : the second equation is just the first one doubled, so the system has infinitely many solutions. Change the second equation to and the last ratio becomes , so now there is no solution.

Three Ways to Solve a System

Once you know a system has exactly one solution, there are three standard ways to find it. Each has its own lesson in this chapter:

Larger systems, with three or more variables, are usually handled with elimination or with matrices. To check your working with any of these methods, the system of equations calculator shows every step.

Writing a System With a Given Solution

Teachers often turn the question around: write a system of equations whose solution is . The trick is to choose the left-hand sides freely, then let the point decide the right-hand sides.

Step 1 — pick any two left-hand sides that are not multiples of each other:

Step 2 — substitute , to find each right-hand side:

Step 3 — write the system:

Because the two left-hand sides have different slopes, the lines cross exactly once, at . Had you picked and , the lines would be parallel or identical, and would not be the only solution.

Worked Example

Gym A charges a joining fee plus a month. Gym B charges to join plus a month. After how many months do the two gyms cost the same, and which is cheaper after that?

Step 1 — define the variables: let be the number of months and the total cost in dollars.

Step 2 — write one equation per gym:

Step 3 — check the number of solutions: the slopes and are different, so there is exactly one solution.

Step 4 — both expressions equal , so set them equal to each other:

Step 5 — find from either equation:

Step 6 — check in the other equation:

After 6 months both gyms have cost . Gym B is cheaper before that point, and Gym A is cheaper after it, because its monthly fee is lower.

Step 4 is a small preview of the substitution method. When both equations are already solved for , you can always set them equal.

Common Mistakes to Avoid

• Checking only one equation. A pair that satisfies one equation is a point on one line. It must satisfy every equation to be a solution.
• Swapping and . The pair means , . The pair is a different point and fails the second equation above.
• Thinking “no solution” means you made a mistake. If the algebra ends in a false statement like , the lines are parallel and the correct answer is “no solution”.
• Answering ”” with ””. A true statement with no variable left means infinitely many solutions, not a solution of zero.
• Comparing slopes before rearranging. In the slope is , not . Solve for first.

Practice Problems

Try each one before opening the answer.

Problem 1. Is a solution of the system and ?

Problem 2. Is a solution of the system and ?

Problem 3. How many solutions does the system and have?

Problem 4. How many solutions does the system and have?

Problem 5. Solve the system and .

Problem 6. Write a system of two linear equations whose only solution is , different from the one in the lesson.

You now know what a solution is and how many to expect. The next step is finding it reliably, and the first method most people learn is solving by substitution, which builds directly on Problem 5.