Linear Equation Solver

1. Pick one variable for an equation like 5x - 7 = 2x + 8, or pick system of two equations for a pair of equations in x and y.
2. Type the number in front of each variable and each plain number. Use a minus sign for negatives, and type 0 for any term that is missing.
3. Read the answer at the top. The cards below show the steps, or the determinant for a system, plus a check that puts the answer back into the equation.
4. Press Copy report or Save to keep the result.

For example, 5x - 7 = 2x + 8 is entered as a = 5, b = -7, c = 2, and d = 8.

A linear equation is an equation where every variable has a power of 1. You will not see x squared, a square root of x, or x under a fraction bar. The graph of a linear equation in two variables is a straight line.

The equations 2x + 3 = 11 and 4x - y = 10 are linear. The equations x^2 = 9 and 1/x = 4 are not, because x has a power other than 1.

A linear equation in one variable has exactly one answer, no answer, or every number as an answer. There are no other options, which makes these equations easy to check.

The goal is to get x alone on one side. Whatever you do to one side, you must also do to the other side. That keeps the equation balanced.

Worked Example: 5x - 7 = 2x + 8

1. Collect the x terms. Subtract 2x from both sides: 3x - 7 = 8.
2. Isolate the x term. Add 7 to both sides: 3x = 15.
3. Divide both sides by 3: x = 5.
4. Check. The left side is 5(5) - 7 = 18. The right side is 2(5) + 8 = 18. Both sides match.

When the Answer Is a Fraction

Take 3x + 2 = 7. Subtract 2 to get 3x = 5. Then divide by 3 to get x = 5/3. As a decimal, that is about 1.666667. The fraction is the exact answer, so keep it when you check your work.

A system is two equations that share the same two variables. The solution is the point where their lines cross.

Substitution

Solve one equation for one variable, then plug it into the other. For x + y = 10 and x - y = 2, the first gives y = 10 - x. Then x - (10 - x) = 2, so 2x - 10 = 2 and x = 6. That makes y = 10 - 6 = 4.

Elimination

Add or subtract the equations so one variable cancels. Adding x + y = 10 and x - y = 2 gives 2x = 12, so x = 6 again. Elimination is often faster when the coefficients already line up.

Cramer's Rule

This calculator uses Cramer's rule, a shortcut based on determinants. It gives the same answer as the other two methods.

For 2x + 3y = 12 and 4x - y = 10: D = 2(-1) - 4(3) = -14. Dx = 12(-1) - 10(3) = -42, and Dy = 2(10) - 4(12) = -28. So x = -42 / -14 = 3 and y = -28 / -14 = 2.

Want to see the lines themselves? Our slope calculator finds the slope of each one.

Sometimes the x terms cancel completely. What is left tells you which case you have.

Systems work the same way. When the determinant D is 0, the two lines have the same slope. If one equation is just a multiple of the other, they are the same line and there are infinitely many solutions. If only the left sides match, the lines are parallel and never meet.

Example: 2x + 4y = 8 and x + 2y = 5. Doubling the second gives 2x + 4y = 10. The left sides match but 8 is not 10, so there is no solution.

1. Sign Errors When Moving Terms

When a term moves to the other side, its sign flips. From 5x - 7 = 8, adding 7 gives 5x = 15. Writing 5x = 1 means the 7 was subtracted instead of added.

2. Not Distributing a Negative

A minus sign in front of brackets changes every term inside. So -(x - 4) is -x + 4, not -x - 4. Missing the second sign is one of the most common algebra slips.

3. Dividing Only One Side, or Only One Term

From 3x = 12 + 6, you must divide the whole right side by 3. The answer is x = 18 / 3 = 6. Dividing only the 12 gives 4 + 6 = 10, which is wrong.

4. Skipping the Check

Put your answer back into the original equation. If both sides do not match, there is an error somewhere. The calculator does this check for you.

A linear equation is an equation where each variable appears only to the first power. There are no squares, roots, or variables in a denominator. Its graph is a straight line, which is where the name comes from.

Move every x term to one side and every plain number to the other side. Then divide both sides by the number in front of x. For 5x - 7 = 2x + 8, subtract 2x and add 7 to get 3x = 15, so x = 5.

It means no value of x can make both sides equal. The x terms cancel out and leave a false statement, such as 1 = 4. The equation 3x + 1 = 3x + 4 is an example.

It means every value of x works. The two sides are the same expression, so the x terms cancel and leave a true statement such as 6 = 6. This kind of equation is called an identity.

Cramer's rule solves a system with determinants. For two equations, D = a1 b2 - a2 b1, and then x = Dx / D and y = Dy / D. It only works when D is not zero.

Use substitution when one variable already has a coefficient of 1 or -1, because it is easy to isolate. Use elimination when the coefficients line up so that adding or subtracting the equations cancels a variable. Both give the same answer.

Many linear equations have answers that are not whole numbers. A fraction such as 5/3 is exact, while the decimal 1.666667 is rounded. The calculator shows both so you can use whichever your class or task needs.

• OpenStax. Elementary Algebra 2e, section 2.3: Solve Equations with Variables and Constants on Both Sides (one-variable solving steps).
• OpenStax. College Algebra 2e, section 7.1: Systems of Linear Equations: Two Variables (substitution, elimination, and inconsistent or dependent systems).
• Dawkins, P. Solving Linear Equations. Paul's Online Notes, Lamar University (worked examples and checking solutions).
• Weisstein, E. W. Linear Equation. MathWorld, Wolfram Research (definition of a linear equation).