Pre-Algebra
Expressions and equations
Systems of linear equations
20 practice questions
0 video lessons
Theory + worked examples
Systems of Linear Equations
Common Core Pre-Algebra • Standard 8.EE.8 • Expressions & Equations
Systems of Linear Equations is a topic in Expressions & Equations in the Common Core State Standards. It is aligned to Standard 8.EE.8, which requires students to solve systems of two linear equations, including by graphing.
A system of linear equations is solved by the point that satisfies both equations — the intersection of their graphs.
Theory
A system of linear equations is two or more equations in the same variables; the solution satisfies all of them.
On a graph, the solution is where the lines intersect.
Two lines meet at the solution.
Solving a system.
The idea:
\[\text{solution} = \text{intersection of the graphs}\]
Substitution and elimination solve systems algebraically.
How to solve a system
- Graph both lines and read the intersection, or
- substitute one equation into the other, or
- eliminate a variable by adding equations.
- Check the point in both equations.
Example 1 — By graphing
Where do \(y=x+1\) and \(y=-x+5\) meet?
Solution
Set equal: \(x+1=-x+5\).
| \(2x\) | \(=\) | \(4\) |
| \((x,y)\) | \(=\) | \((2,3)\) |
Example 2 — Substitution
Solve \(y=2x\), \(x+y=6\).
Solution
Substitute \(2x\) for \(y\).
| \(x+2x\) | \(=\) | \(6\) |
| \(x\) | \(=\) | \(2\) |
Example 3 — Check
Is \((2,4)\) a solution of \(y=2x\) and \(x+y=6\)?
Solution
\(4=2(2)\) ✓ and \(2+4=6\) ✓.
Example 4 — Meaning
What does the intersection represent?
Solution
The values satisfying both equations at once.
Common pitfalls
The solution must satisfy both equations.
Parallel lines give no solution.
Read the intersection carefully from a graph.
Frequently asked questions
What is a system of equations?
Two or more equations in the same variables.
What is the solution?
The values satisfying all equations — the intersection.
How can I solve one?
Graphing, substitution, or elimination.
What if the lines are parallel?
No solution.
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