Analyze and solve pairs of simultaneous linear equations.
Official wording from the Common Core State Standards for Mathematics (Β© 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers). View on thecorestandards.org
A system of two linear equations asks for an ordered pair that makes both equations true at the same moment. Graphically, each equation is a line, and a point that satisfies both must sit on both lines, which is exactly where they cross. Eighth graders learn this link first, because it explains what an algebraic answer means: the solution (3, 2) is the intersection point of the two graphs.
Students then solve systems algebraically, typically by substitution (for example replacing y with 2x - 1 in the second equation) or by adding or subtracting equations to eliminate a variable. They also solve some systems by inspection. Two lines with the same slope and different intercepts never meet, so the system has no solution; equations that describe the same line have infinitely many. Real-world problems, such as comparing two phone plans or finding when two tanks hold the same amount, give the solution a meaning in context.
Students find x = 3 and stop. The solution to a system is an ordered pair, so they must substitute back to find y as well.
A pair can satisfy one equation by chance. A true solution must make both equations true, so check it in each one.
Lines with equal slopes and different intercepts stay the same distance apart forever, so the system has no solution at all.
A sketch can suggest (2, 3) when the exact answer is (2.2, 3.4). Graphs give estimates; algebra confirms the exact values.
Solve the system y = 2x - 1 and 3x + y = 14.
Answer: The solution is (3, 5), where the two lines intersect.
Begin with a context where students can guess and check, such as two gyms with different joining fees and monthly costs, and record the totals in a table month by month. The month where the totals match becomes the intersection on a graph and then the solution of the equations, so all three views tell the same story.
Graphing technology is useful for checking but should not replace understanding. Assessment items may ask for the number of solutions from equations alone, so practice rewriting both in y = mx + b form and comparing slopes and intercepts.
Original questions written for this standard. Choose an option or type your answer, then press Check. Every question has a worked explanation.
Answer: 2 (also accepted: x = 2, x=2)
Set the expressions equal: 3x = x + 4, so 2x = 4 and x = 2. Then y = 6, giving the solution (2, 6).
Answer: A) None
Both lines have slope 2 but different intercepts, so they are parallel and never intersect. There is no solution.
Answer: B) (7, 3)
Adding the equations gives 2x = 14, so x = 7. Then y = 10 - 7 = 3. Check: 7 - 3 = 4.
Answer: 20/3
Set 20 + 5m = 8m. Subtract 5m: 20 = 3m, so m = 20/3, about 6.7 months. Plan B is cheaper before that point and Plan A after it.
Answer: D) It is the only point that lies on both lines, so it satisfies both equations
Every point on a line satisfies that line's equation. The intersection lies on both lines, so it satisfies both equations at once.
Answer: 3 (also accepted: x = 3, x=3)
Substitute y = 2: 2x + 6 = 12, so 2x = 6 and x = 3. The solution is (3, 2).
Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
A full lesson with slides, activities and an exit ticket on systems of linear equations, pitched to grade 8 and editable in PowerPoint or Google Slides.
Make a lesson βA printable, differentiated worksheet on 8.EE.C.8 with an answer key, ready in about a minute.
Make a worksheet βTurn systems of linear equations into a quiz students answer online that marks itself, with a class summary for you.
Build a test βThe standard expects graphing for estimates, algebraic solving and solving simple cases by inspection. Substitution is the most common algebraic method at this grade, with elimination often introduced too.
The two equations describe the same line, so every point on that line satisfies both equations.