Solving Systems of Linear Equations: Step-by-Step Guide
A system of linear equations is a set of two (or more) equations that use the same variables. The solution is the pair of numbers that makes all the equations true at the same time. Two main ways to solve a system are substitution (solve one equation for one variable and plug it into the other) and elimination (add or subtract the equations to cancel out one variable).
Math Racer skill guide · Algebra 1 · Grade 9
How to solve it, step by step
- Choose a method. Use substitution if one equation already has a variable by itself (or is easy to rearrange). Use elimination if the numbers in front of a variable are the same or opposites.\[\begin{cases} a_{1}x+b_{1}y = c_{1} \\ a_{2}x+b_{2}y = c_{2} \end{cases}\]
- Get rid of one variable. For elimination: multiply the equations if needed so one variable has opposite numbers in front of it, then add the equations. For substitution: solve one equation for a variable, then put that into the other equation.\[(a_{1}x+b_{1}y) + (a_{2}x+b_{2}y) = c_{1}+c_{2}\]
- Solve the one-variable equation. You should now have one equation with only one variable. Solve it.\[kx = m \;\Rightarrow\; x = m/k\]
- Find the other variable. Put your answer into one of the original equations and solve for the other variable.\[y = \frac{c_{1} - a_{1}x}{b_{1}}\]
- Check in BOTH equations. Put both values into each of the original equations. They must both be true.\[(x,y) \text{ satisfies equation 1 AND equation 2}\]
Worked example
Solve the system: 2x + y = 7 and 3x - y = 8
Step 1: We will use elimination, because the y terms (+y and -y) are already opposites.
Step 2: Add the two equations: (2x + y) + (3x - y) = 7 + 8, which gives 5x = 15.
Step 3: Solve for x: x = 3.
Step 4: Put x = 3 into 2x + y = 7: 2(3) + y = 7, so 6 + y = 7, and y = 1.
Step 5: The solution is x = 3 and y = 1, or the point (3, 1).
Check: Equation 1: 2(3) + 1 = 7. True. Equation 2: 3(3) - 1 = 8. True. Both work.
Tips and common mistakes
- Pick the method that is easiest for the problem. If the numbers already line up, elimination is faster.
- When you substitute, put the whole expression in parentheses to avoid sign mistakes.
- If everything cancels and you get something true like 0 = 0, the equations describe the same line and there are infinitely many solutions.
- If everything cancels and you get something false like 0 = 5, there is no solution.
- Always check your answer in BOTH original equations. One working does not mean the other will.
- On a graph, the solution is the point where the two lines cross. Parallel lines never cross, so there is no solution. If the lines are the same, there are infinitely many.