CCSS.Math.Content.HSA-REI.C.5
The standard
Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
Common Core State Standards for Mathematics
What this standard means
Students need to understand why adding a multiple of one equation to another does not change the solution set of a linear system. They should connect this move to elimination, not just follow steps. If a point satisfies both original equations, it must also satisfy the new combined equation.
Mastery looks like explaining the logic with variables, numbers, or a graph. Students can say why the intersection point stays the same after the replacement. Common trouble spots are treating the move like magic, changing both equations at once, or forgetting that one original equation must remain in the system.
Ways to teach it
- Give pairs two line equations, have them graph both, replace one equation using elimination, then graph the new line to compare intersections.
- Ask students to write: Why does adding three times one true equation to another true equation still give a true equation?
- Show a worked replacement step and ask students to circle whether the solution set changed, then justify in one sentence.
- Connect to balancing receipts: if two totals are both correct, adding three copies of one receipt to another keeps the shared item values consistent.
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Related standards
- CCSS.Math.Content.8.EE.C.8
Analyze and solve pairs of simultaneous linear equations.
- CCSS.Math.Content.HSA-REI.C.7
Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.
- CCSS.Math.Content.HSA-REI.C
Solve systems of equations
- CCSS.Math.Content.8.EE.C.8a
Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersect...