Grade 8 · Locate simultaneous solutions · 462 of 750
Locate simultaneous solutions · practice 53
Solve the simultaneous equations y = -3x and y = 1.5x + (-1). Classify the number of solutions. Follow the calculation, then test the quantities in the animated example.
Try the question, use a hint when you need one, and compare your reasoning with the worked solution. The animation starts with this chapter’s values; changing its controls explores a new case.
01 · Make a prediction
Your practice question
Solve the simultaneous equations y = -3x and y = 1.5x + (-1). Classify the number of solutions.
Use this scratch space or work on paper. Your notes stay on this page and clear when you leave. Answers are for self-checking; they are not automatically graded.
A starting point
At an intersection, the two expressions for y are equal.
Work through the reasoning
Step 1
Translate the givens
Equate outputs: -3x = 1.5x + (-1). Subtract 1.5x to obtain (-4.5)x = -1.
Step 2
Calculate with the model
Divide by -4.5: x = -1/-4.5 ≈ 0.22222. Then y = -3x ≈ -0.66667.
Step 3
Check and interpret
The difference of the two outputs is (-4.5) × (-1/-4.5) − (-1) = 0, so the exact point satisfies both equations.
The answer
x = -1/-4.5 ≈ 0.22222; y = -3 × (-1/-4.5) ≈ -0.66667.
Compare the method as well as the result. A different valid method may reach the same answer. Keep exact values until the last step when the question asks for rounding.
02 · Explore the model
See the mathematical relationship move.
Use Play, Pause, and the timeline to inspect the construction. Reset restores the question’s original settings. The displayed assumptions describe where this model applies.
Watch the relationship
PausedHD animation studio
From experiment to screen.
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The mathematical idea
A shared solution satisfies two line equations at the same input and output. Different slopes yield one crossing, equal slopes with different intercepts yield none, and identical rules share infinitely many points. Two equations alone do not guarantee a unique solution. Starting quantities: First slope a = -3; Second slope b = 1.5; Second intercept c = -1.
y = ax and y = bx+c; (a−b)x = c
03 · Reflect and transfer
Explain what changes and why.
What happens to the number of solutions when slopes agree but intercepts do not?
This is a distinct guided scenario using a reusable mathematical model. Similar-looking diagrams can represent different given values and conclusions; they are not different mathematical theories.