This grade 8 systems of equations lesson explains systems of linear equations in one printable page, with definitions, a method and a worked example.
A system of linear equations is two linear equations with the same variables, and its solution is the ordered pair that makes both equations true. On a graph, the solution is where the lines cross. You can find it by graphing, substitution or elimination, and a system can have one solution, none or infinitely many.
Grade 8 Systems of Equations Lesson: key facts
- Standards 8.EE.C.8a, 8.EE.C.8b and 8.EE.C.8c: solve systems of two linear equations in two variables and use them in real-world problems.
- The solution of a system is the point where the two graphs intersect.
- Substitution replaces one variable with an expression; elimination adds or subtracts equations to cancel a variable.
- Parallel lines (same slope, different intercepts) mean no solution; the same line twice means infinitely many.
- Always check the ordered pair in both original equations.
What is a system of linear equations?
A system of linear equations is a set of two or more linear equations that use the same variables. A solution is an ordered pair \((x,y)\) that makes every equation true at the same time. On a graph, it is the point where the lines cross.
System: two equations solved together, such as \(y=2x-1\) and \(y=-x+5\).
Solution: the ordered pair that satisfies both equations.
Elimination: adding or subtracting equations to remove one variable.
How do you solve a system by graphing?
- Graph both lines on one coordinate plane.
- Find the point where they cross.
- Substitute that point into both equations to check it.
In the graph above, \(2(2)-1=3\) and \(-2+5=3\), so \((2,3)\) works in both equations.
How do you solve a system by substitution?
Replace a variable with an expression from the other equation. To solve \(y=2x-1\) and \(x+y=8\), put \(2x-1\) in place of \(y\): \(x+(2x-1)=8\). Then \(3x=9\), so \(x=3\) and \(y=2(3)-1=5\).
How do you solve a system by elimination?
Add or subtract the equations so one variable drops out. For \(x+y=11\) and \(x-y=3\), adding gives \(2x=14\), so \(x=7\). Then \(7+y=11\) gives \(y=4\). If no variable cancels, multiply one or both equations by a number first.
Can a system have no solution or infinitely many?
Yes. Lines with the same slope and different intercepts are parallel and never meet, so there is no solution. By inspection, \(3x+2y=5\) and \(3x+2y=6\) cannot both hold, because the same expression cannot equal 5 and 6. If both equations describe the same line, such as \(y=x+1\) and \(2y=2x+2\), there are infinitely many solutions. Otherwise the lines cross exactly once.
How do you use a system in a word problem?
Noah buys 6 items at the school store for $14. Pens cost $1 and notebooks cost $3. Let \(p\) be pens and \(n\) be notebooks. Then \(p+n=6\) and \(p+3n=14\). Subtract the first from the second: \(2n=8\), so \(n=4\) and \(p=2\).
Systems of Linear Equations: frequently asked questions
What does the solution of a system of equations mean?
It is the ordered pair that makes both equations true. On a graph it is the intersection point of the two lines. For \(y=x+1\) and \(y=-x+5\), the solution is \((2,3)\).
When should I use substitution instead of elimination?
Use substitution when one equation already has a variable alone, such as \(y=3x\). Use elimination when both equations are in the form \(Ax+By=C\) and a variable can cancel.
How can I tell that a system has no solution without graphing?
Compare slopes and intercepts. If the slopes are equal and the intercepts differ, the lines are parallel. For example, \(y=2x+1\) and \(y=2x-3\) never meet.
What if elimination ends with a true statement like 0 = 0?
Both equations describe the same line, so every point on it is a solution. The system has infinitely many solutions.
How do I check my answer to a system?
Substitute both \(x\) and \(y\) into each original equation. If both equations are true, the pair is correct. Checking only one equation is not enough.
How to use this grade 8 systems of equations lesson
Use this grade 8 systems of equations lesson before the worksheet, then check what you learned with the quiz.
See every chapter on the Grade 8 math page.