How many solutions can two equations have?

You need coordinates and substitution for this lesson. In $y=mx+b$, $m$ is the slope and $b$ is where the line crosses the vertical axis.

In the previous lesson, you adjusted coefficients to reach a target. Now we will describe the allowed choices with equations.

Consider two rules:

$$y=x,\qquad y=-x+2.$$

Find a pair $(x,y)$ that obeys both. Try $(0,0)$: it works in the first rule, but the second asks for $y=2$. Try $(1,1)$ before continuing.

Each line contains all the points that obey one rule. Where would a point that obeys both have to sit?

Keep a record of your investigation

Use $y=x$ and $y=-x+2$ as the reference pair. Write down the predicted number of shared points before each change. First change only the second slope, keeping both intercepts fixed. Then keep the equal slopes fixed and change only the second intercept. Record the equations and shared points in each case. Explain how the changed coefficient accounts for the result.

Predict, then move the lines

The explorer starts with the two rules above. Before selecting Same slope, predict whether the lines will share one point, no points, or many points.

Keep the equal slopes fixed. Before selecting Same line, predict what changes when the second intercept also matches the first. Then test and record the result.

Can both equations be true?Reset

Equation 1 (solid)−x + y = 0

Equation 2 (dashed)x + y = 2

Second line's slope

-1

Second line's intercept

2

Crossing linesSame slopeSame line

One solution: (1, 1).

Pause and describe each arrangement in words. Explain why matching slopes alone is different from matching both slope and intercept.

Compare your explanation

The starting lines cross at $(1,1)$. With the same slope but different intercepts, the lines stay apart. When both slope and intercept match, the two drawings occupy the same line: every point on it satisfies both equations.

A solution must satisfy every equation

A system of equations asks us to satisfy several equations simultaneously. A solution is a choice of values that makes all of them true.

For the starting pair, substitute $y=x$ into $y=-x+2$:

$$x=-x+2\quad\Longrightarrow\quad 2x=2\quad\Longrightarrow\quad x=1.$$

Then $y=1$. Substitution confirms that $(1,1)$ works in both equations.

For the parallel case, the rules are $y=x$ and $y=x+2$. Substitution would require $x=x+2$, or $0=2$. No real number can make that true.

For the overlapping case, both rules say $y=x$. Any real number $t$ gives a solution $(t,t)$, including $(0,0)$, $(3,3)$, and $(-7,-7)$.

Why there cannot be exactly two shared points

Two distinct points determine a straight line. If both lines passed through the same two distinct points, they would be the same line and share all its points.

So for two straight lines in the plane, the possibilities are one, none, or infinitely many shared points. This argument concerns straight lines; a line and a circle can meet twice.

A system with at least one solution is consistent. One with no solution is inconsistent.

Different writing, same restriction

Before calculating, decide whether these equations describe different lines:

$$x+y=4,\qquad 2x+2y=8.$$

Dividing the second equation by 2 recovers the first. There are two written equations but just one restriction. For every real $x$, choosing $y=4-x$ satisfies both.

When constraints describe measurements or model requirements, counting them is not enough: some may repeat information, and others may conflict.

Design your own comparison

Keep the first line $y=x$ fixed and choose your own second line in the explorer so that they share exactly one point. Write down their equations. Design a change to one coefficient that makes them parallel and distinct, then a second change to one coefficient that makes them coincide. Predict before each change and record the result.

Explain each verdict by substitution as well as by the picture. If your first choice cannot produce all three cases in that order, explain why and revise it.

Write a short conclusion: what you predicted, what you observed, and how the evidence changed or supported your explanation.

Check the reasoning in your investigation

Your crossing pair needs unequal slopes. To get distinct parallel lines by changing one slope, the intercepts must already differ. Once the slopes match, matching the intercepts makes the equations identical. A crossing outside the visible graph is still a shared point, so the equations are the decisive check.

Check your understanding

These questions use new points and equations. Select an answer, then use Check answer to compare your reasoning.

Question 1 of 30 checked · 0 correct

Does (2, 2) solve both y = x and y = −x + 2?

Yes, because x and y are equal.No. It satisfies only the first equation.No. It satisfies neither equation.

Check answer

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