Two equations, one answer: what a solution of a system is and why it must pass both equations; solving by graphing, by substitution and by elimination, and why adding a multiple of one equation to the other is allowed; one solution, none, or infinitely many, read from the slopes or from 0 = 6 and 0 = 0; choosing the method the system hands you. Then the modeling side: tickets, coins, mixtures and a boat in a current, with the answer stated in units. Finally linear inequalities in two variables as half-planes, and a system of inequalities as the region where they overlap. Study cards, hints, a practice quiz at three levels, and four workshop activities that check themselves.
Every point on this line makes the first equation true. There are infinitely many of them.
A system's answer is the point that sits on BOTH lines — a pair, (x, y), not a single number. That is exactly why you check it in both equations: being on one line proves nothing at all.
What people get wrong
⚠️People often think…
If the pair works in one equation, it solves the system.
Infinitely many points sit on any one line. (0, 1) is on y = 2x + 1 and that says nothing about the other equation. A solution has to pass BOTH, which takes two substitutions and about fifteen seconds.
Two equations means two checks. Always.
⚠️People often think…
To eliminate, just subtract the second equation.
Subtracting means changing EVERY sign in the second equation, including the constant — and that is where it goes wrong. Multiply the whole equation by −1 and add instead. Adding is much harder to fumble.
Multiply by −1 and add. Never subtract.
⚠️People often think…
Multiplying an equation only changes the terms with variables.
It changes everything, constant included. −2 times (x + y = 6) is −2x − 2y = −12, not −2x − 2y = 6. Leave the constant alone and you have written down a different line entirely.
The multiplier crosses the equals sign too.
⚠️People often think…
Both equations in a word problem count the same thing.
Usually one counts things and the other counts money. Tickets: a + s = 200 counts PEOPLE; 8a + 5s = 1300 counts DOLLARS. Mixing the two into one equation is the most common way these problems fall apart.
Label each equation with what it counts before you solve.
Find the pair, check the pair
1Watch one
Is (2, 3) the solution of x + y = 5 and x − y = −1?
First equation: 2 + 3 = 5. True.
That alone is not enough.
Second equation: 2 − 3 = −1. Also true.
Both hold, so (2, 3) is the solution.
2Do one with me
Solve by elimination: x + y = 10 and x − y = 4.
Add the two equations. The y's cancel: 2x =
So x =
Put it back in the first equation: y =
💬One sentence, then you move on
Why isn't finding x the end of the problem?
3Try one
You solve a system and both variables vanish, leaving 0 = 6. How many solutions does it have?
I want a hint first
It is not x = 0. Both variables are gone, so read what is left: false means no solution, true means infinitely many.
💬Last one — then you're done here
Why do you have to check the pair in both equations?
Where this goes
Where this lives
Two plans that cost the same at some number of units — phone plans, gym memberships, rental cars. The crossing point is the number where the cheaper option switches.
What this feeds
Next unit is exponents, polynomials and factoring — the tools you need before quadratics.
Name two options you have chosen between where the better deal depended on how much you used.
One card at a time — tap “Show me” to check yourself, then Next. Start at Foundation; when those feel easy, climb.
Helpful Hints
🪜 Solving a system — the routine
Look at how the system is written and pick the method it hands you → get ONE equation in ONE variable (substitute an isolated expression, or add a multiple of one equation to the other so a variable cancels) → solve it → substitute back to find the other variable → write the answer as an ordered pair (x, y) → check the pair in BOTH original equations → in a story, say what the pair means with units.
Pick, reduce, solve, back-substitute, check both.
💡 The whole unit in one idea
Each equation is a line — a whole street of points that satisfy it. A system asks for the point that lives on BOTH streets: their crossing. Graphing finds it by eye, substitution and elimination find it exactly, and elimination is legal because adding two true equations makes a third true equation through the same point. Change the equals signs to inequalities and each line becomes a half-plane, and the crossing becomes the region where the half-planes overlap.
⚠️ Traps the test loves
Checking the pair in only one equation. (0, 1) is on y = 2x + 1 but that says nothing about the system.
Subtracting equations and forgetting to change EVERY sign of the second one. Multiply by −1 and add instead.
Multiplying an equation to line up coefficients but leaving the constant alone: −2(x + y = 6) is −2x − 2y = −12.
Writing x = 0 when both variables vanish. 0 = 6 means no solution; 0 = 0 means infinitely many.
Mixing a count with a value in one equation. Tickets: a + s = 200 counts people; 8a + 5s = 1300 counts dollars.
Shading the side that contains the test point without actually testing it. Substitute (0, 0) and read true or false.
📋 Which method, and what the algebra tells you — the chart
The system looks like…
Use…
First move
y = 2x + 1 and y = −x + 4
graphing, or set them equal
2x + 1 = −x + 4
y = 3x − 1 and 2x + y = 9
substitution
2x + (3x − 1) = 9
2x + 3y = 12 and 4x − 3y = 6
elimination — add
6x = 18
3x + 2y = 16 and x + y = 6
elimination — multiply, then add
−2(x + y = 6), then add
the algebra ends in 0 = 6
stop
no solution — parallel lines
the algebra ends in 0 = 0
stop
infinitely many — the same line
🎯 How the test will ask
A pair and a system — “is it a solution?” Substitute into both; two check marks or it is not.
Two lines on a grid — “what is the solution?” Read the crossing as (x, y), then verify it in both equations.
A system in standard form — “solve by elimination.” Match or oppose a coefficient, add, back-substitute.
One equation with y alone — “solve by substitution.” Parentheses around the substituted expression.
Three systems — “one solution, none, or infinitely many?” Compare slopes, then intercepts.
Tickets, coins, a mixture, a boat in a current — “write a system and solve.” One equation for count or volume, one for value.
Two inequalities — “graph the solution region” or “is this point in it?” Solid or dashed, test (0, 0), find the overlap.
✅ Before the test, can you…
Check whether an ordered pair solves a system, and say why one equation is not enough?
Solve a system by graphing, and say when the graph is only an estimate?
Solve by substitution and by elimination, including systems that need a multiplication first?
Explain why adding a multiple of one equation to the other keeps the same solutions?
Recognize no solution and infinitely many solutions, from the slopes and from 0 = 6 or 0 = 0?
Write and solve a system for tickets, coins, a mixture or a rate problem, and state the answer with units?
Graph a linear inequality and a system of two, and test whether a point is in the region?
Pick your level
Look back at anything you missed — the hint that appeared is exactly what to reread tonight.
How sure did you feel?
Workshop
Work like a mathematician: put an elimination solution in order with the reason for every step, match each system to the method it hands you, draw two lines on a four-quadrant plane with your fingertip and read where they meet, then settle a ticket-sales argument with both equations. Every activity checks itself, and hints are free.
Your practice never leaves this device. There is no account and no sign-in. Your work is saved in this browser only, and you can erase it whenever you want.
Your practice record — saved on this device
This is your record of the module on screen — it stays here and goes nowhere. Independent means you got it right on the first tap; supported means you got it after the explain-and-retry, or marked ‘I had it’ on a revealed answer. Both count, and neither is a grade. If your teacher asks, copy the row or show them this screen.
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