Systems of Linear Equations
Lessons in this chapter
About Systems of Linear Equations
Systems of Linear Equations takes two linear equations at once and finds the single point (if any) where both are true.
Graphing Systems of Linear Equations finds a system's solution visually: the point where the two lines intersect, before Algebraic Methods of Solving Systems of Linear Equations introduces substitution and elimination, two ways to find that same point without graphing, especially useful when the intersection isn't a neat pair of numbers.
Modeling using Systems of Linear Equations applies both methods to real two-unknown situations, like comparing two pricing plans to find when they cost the same, turning a word problem into two equations and solving for where they agree.
The three lessons, Graphing Systems of Linear Equations, Algebraic Methods of Solving Systems of Linear Equations and Modeling using Systems of Linear Equations, are 8th grade math worksheets worked in the page. Each answer is checked as it is given, so a pair of values that satisfies only one of the two equations is caught rather than accepted. All three are free to try, with no login needed for your first 3.
Frequently asked questions
Graphing Systems of Linear Equations comes first, with the solution read as the point where two lines cross, and the two special cases of parallel lines with no solution and identical lines with infinitely many. Algebraic Methods of Solving Systems of Linear Equations then covers substitution and elimination for exact answers a graph cannot give, and Modeling using Systems of Linear Equations sets up both equations from a word problem before solving. This sits in the Common Core domain Expressions and Equations (8.EE).
A system of linear equations is two (or more) linear equations considered together, and "solving" the system means finding the point (x, y) that makes both equations true at once.
Graph both lines on the same coordinate plane: the point where they cross is the system’s solution, since it’s the one point that lies on both lines at once.
Solve one equation for one variable, then substitute that expression into the other equation, reducing the problem to a single one-variable equation to solve, then use that answer to find the other variable.
Add or subtract the two equations (after matching coefficients, if needed) so that one variable cancels out entirely, leaving a single-variable equation to solve.
A system has no solution when the two lines are parallel (same slope, different intercept) and never cross, and infinitely many solutions when the two equations describe the exact same line.