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Two lines, and what their crossing point means.
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In this lesson you solve two linear equations at once, finding the pair of values that satisfies both. Graphically it is where the two lines cross, which immediately explains the strange cases: parallel lines never cross, so there is no solution, and identical lines cross everywhere, so there are infinitely many.
You can graph a line from an equation like $y = 2x + 3$: the $3$ is where it crosses the $y$-axis and the $2$ is its slope. You know that every point on the line is a pair $(x, y)$ that makes the equation true, and every point off the line makes it false. You can solve equations with the variable on both sides, such as $5x - 4 = 2x + 11$, and you can substitute a number for a letter and work out the result. This lesson puts two equations together. Instead of one unknown, there are two, $x$ and $y$, and you need a pair of values that makes both equations true at the same time.
| Term | What it means |
|---|---|
| System of equations | Two (or more) equations about the same unknowns, to be solved together. |
| Solution of a system | A pair $(x, y)$ that makes every equation in the system true. |
| Point of intersection | The point where two graphs cross. For two lines it is the solution of their system. |
| Substitution | Replacing a letter in one equation with an expression the other equation says it equals. |
| Elimination | Adding or subtracting the equations so that one letter cancels out. |
| Parallel lines | Lines with the same slope that never meet. Their system has no solution. |
| Coincident lines | Two equations that describe the same line. Their system has infinitely many solutions. |
One linear equation in two letters, such as $x + y = 10$, has many solutions: $(1, 9)$, $(4, 6)$, $(7.5, 2.5)$ and endless others. Its graph is a line, and every point on the line is a solution.
Add a second equation, $y = x + 4$, and you ask for something stricter: a pair that makes both true. That is a system of two equations. Its solution is a point that lies on both lines, which is the point where they cross. Here that is $(3, 7)$. Check it: $3 + 7 = 10$, and $3 + 4 = 7$. Both equations are true, so $(3, 7)$ solves the system.
You can find that point three ways. Graphing: draw both lines and read off where they cross. Substitution: use one equation to replace a letter in the other, leaving one equation in one letter. Elimination: add or subtract the equations so one letter cancels. Each way turns a problem with two unknowns into a problem with one, which you already know how to solve.
Because the solution is a crossing point, the picture tells you how many solutions to expect. Two lines with different slopes cross exactly once, so there is one solution. Two parallel lines never cross, so there is no solution. Two equations that describe the same line share every point, so there are infinitely many solutions.
Another way: picture
Draw $x + y = 10$ falling from $(0, 10)$ to $(10, 0)$, and $y = x + 4$ rising from $(0, 4)$ with slope $1$. They cross at $(3, 7)$. That point is the only one on both lines.
Another way: table of values
List pairs for each equation. For $x + y = 10$: $(1, 9)$, $(2, 8)$, $(3, 7)$, $(4, 6)$. For $y = x + 4$: $(1, 5)$, $(2, 6)$, $(3, 7)$, $(4, 8)$. The pair in both lists, $(3, 7)$, is the solution.
To solve a system by graphing, draw each line carefully on the same grid. Put each equation in the form $y = mx + b$ first, so you can start at the intercept $b$ and step along with the slope $m$. The solution is the point where the lines cross. Read its coordinates, then check them in both equations, because a graph is only as exact as your drawing.
Graphing shows the whole situation at a glance. You can see at once whether the lines cross, never meet, or lie on top of each other.
Its weakness is precision. If the lines cross at $(2.4, 5.8)$, a hand-drawn graph might suggest $(2, 6)$ or $(2.5, 6)$, and you cannot tell which. That is why the algebra methods matter. They give exact answers, even when the answer is a fraction. A good habit is to use a sketch to predict roughly where the answer should be, and the algebra to find it exactly. If the two disagree badly, something has gone wrong.
Substitution works best when one equation already says what a letter equals, as in $y = 3x - 2$.
Take the system $y = 3x - 2$ and $2x + y = 13$. The first equation says $y$ and $3x - 2$ are the same number. So wherever $y$ appears in the second equation, you may write $3x - 2$ instead. Use brackets:
$$2x + (3x - 2) = 13.$$
Now there is only one letter. Collect: $5x - 2 = 13$, so $5x = 15$ and $x = 3$. Then put $x = 3$ back into the simpler equation: $y = 3 \times 3 - 2 = 7$. The solution is $(3, 7)$.
If both equations are solved for $y$, as in $y = 2x + 5$ and $y = -x + 11$, substitution means setting the two expressions equal: $2x + 5 = -x + 11$. That is an equation with $x$ on both sides, which you solved in the last lesson: $3x = 6$, $x = 2$, and $y = 9$.
The brackets matter when a number multiplies $y$. In $4x - 2y = 6$, putting in $y = 3x - 2$ gives $4x - 2(3x - 2) = 6$, and the $-2$ must multiply both terms: $4x - 6x + 4 = 6$.
Elimination works best when both equations are in the form $ax + by = c$ and one letter has matching or opposite coefficients.
Take $4x + 3y = 26$ and $2x - 3y = 4$. The $y$ terms are $+3y$ and $-3y$. If you add the two equations, left side to left side and right side to right side, they cancel:
$$(4x + 3y) + (2x - 3y) = 26 + 4 \quad\Rightarrow\quad 6x = 30.$$
So $x = 5$. Put it into either equation: $4(5) + 3y = 26$ gives $3y = 6$, so $y = 2$. The solution is $(5, 2)$.
Why may you add two equations? Each equation says two amounts are equal. Adding equal amounts to equal amounts gives equal totals. It is the same idea as adding the same number to both sides, except that the "number" added is written two different ways.
If the matching terms have the same sign, as in $3x + 2y = 12$ and $x + 2y = 8$, subtract instead: $(3x + 2y) - (x + 2y) = 12 - 8$ gives $2x = 4$. Then $x = 2$ and $y = 3$.
Write both equations in the form $y = mx + b$ and compare them.
Different slopes: the lines cross once, so there is one solution. $y = 2x + 1$ and $y = -x + 4$ have slopes $2$ and $-1$.
Same slope, different intercepts: the lines are parallel and never cross, so there is no solution. $y = 3x + 1$ and $y = 3x - 4$ are like two train rails. If you try the algebra anyway, the letters cancel and a false statement is left: $3x + 1 = 3x - 4$ becomes $1 = -4$.
Same slope and same intercept: the two equations describe the same line, so every point on it is a solution: infinitely many. This often hides behind a multiple. $y = 2x - 3$ and $4x - 2y = 6$ look different, but the second rearranges to $-2y = -4x + 6$, which is $y = 2x - 3$. The algebra ends in a true statement such as $0 = 0$.
So the three outcomes you met with one equation, one answer, none or every number, show up again here, and now they have a picture.
Many real problems have two unknowns and two facts about them. Each fact becomes one equation.
A common pattern is one equation that counts things and one that counts their value, whether money, points or weight.
Why the moves are allowed. Substituting replaces a letter with something equal to it, which cannot change whether an equation is true. Adding two true equations gives a true equation. So the pair you find satisfies the original system.
How to check. Put both values into both original equations. The pair must work in each. A pair that works in only one equation is a point on only one of the lines, and it is not the solution.
On March 2, 1962, Wilt Chamberlain scored $100$ points in one NBA game, a record that still stands. There was no three-point line then, so each basket was worth $2$ points and each free throw $1$. He made $64$ scoring shots in all. How many were baskets? Let $b$ be baskets and $f$ free throws. Then $b + f = 64$ and $2b + f = 100$. Subtract the first equation from the second: $b = 36$. Then $f = 64 - 36 = 28$. The box score agrees: $36$ field goals and $28$ free throws.
A student starts a T-shirt printing business. The screen and ink cost $150$ dollars to set up, and each blank shirt costs $6$ dollars. Each printed shirt sells for $16$ dollars. The cost of making $x$ shirts is $y = 150 + 6x$, and the money taken in is $y = 16x$. Where do the lines meet? Set them equal: $16x = 150 + 6x$, so $10x = 150$ and $x = 15$. At $15$ shirts both are $240$ dollars. Below $15$ shirts the business loses money; above it, each shirt adds $10$ dollars of profit. The crossing point is the break-even point.
A jar holds only dimes and quarters: $40$ coins worth $7.00$ dollars in all. How many of each? Work in cents. With $d$ dimes and $q$ quarters, $d + q = 40$ and $10d + 25q = 700$. From the first, $d = 40 - q$. Substitute: $10(40 - q) + 25q = 700$, so $400 + 15q = 700$ and $q = 20$. Then $d = 20$. Check: $20$ dimes make $200$ cents and $20$ quarters make $500$ cents, $700$ cents in all.
Stopping after one letter. Finding $x = 3$ is half the answer. The solution is a pair, such as $(3, 7)$.
Checking in only one equation. The pair must work in both.
Forgetting the brackets when substituting. In $4x - 2y = 6$ with $y = 3x - 2$, write $4x - 2(3x - 2)$, not $4x - 6x - 2$.
Adding equations whose terms do not cancel. Add when the coefficients are opposites; subtract when they are the same.
Calling parallel lines one solution. Equal slopes with different intercepts never meet, so there is no solution.
Solve $y = x - 2$ and $y = -x + 4$. Make a table for the first line.
$(0, -2), \; (1, -1), \; (2, 0), \; (3, 1)$
Each pair makes $y = x - 2$ true.
Make a table for the second line with the same $x$ values.
$(0, 4), \; (1, 3), \; (2, 2), \; (3, 1)$
Each pair makes $y = -x + 4$ true.
Find the pair that is in both tables.
$(3, 1)$
That point is on both lines, so the lines cross there.
Check it in the first equation.
$1 = 3 - 2$
The pair makes the first equation true.
Check it in the second equation.
$1 = -3 + 4$
It works in both, so the solution is $(3, 1)$.
Solve $y = 4x - 1$ and $y = x + 8$. Set the two expressions for $y$ equal.
$4x - 1 = x + 8$
At the solution both equations give the same $y$.
Subtract $x$ from both sides.
$3x - 1 = 8$
Now $x$ is on the left only.
Add $1$ to both sides.
$3x = 9$
Undo the number term.
Divide both sides by $3$.
$x = 3$
That is half of the answer.
Put $x = 3$ into the simpler equation.
$y = 3 + 8 = 11$
Either equation gives $y$; $y = x + 8$ has less arithmetic.
Check the pair in the other equation.
$4(3) - 1 = 11$
It works in both, so the solution is $(3, 11)$.
Solve $3x + 2y = 17$ and $5x - 2y = 7$. Look at the $y$ terms.
$+2y \text{ and } -2y$
They are opposites, so adding the equations will cancel them.
Add the left sides.
$(3x + 2y) + (5x - 2y) = 8x$
$3x + 5x = 8x$ and $2y - 2y = 0$.
Add the right sides and write the new equation.
$8x = 17 + 7 = 24$
Equal amounts added to equal amounts give equal totals.
Divide both sides by $8$.
$x = 3$
Dividing undoes multiplying by $8$.
Substitute $x = 3$ into the first equation.
$9 + 2y = 17$
$3 \times 3 = 9$.
Subtract $9$ from both sides.
$2y = 8$
Get the $y$ term alone.
Divide both sides by $2$.
$y = 4$
The solution is the pair $(3, 4)$.
Check the pair in the second equation.
$5(3) - 2(4) = 15 - 8 = 7$
It matches, so $(3, 4)$ solves both equations.
Subtract $6x$ from both sides of the second equation.
$-2y = -6x + 10$
Get the $y$ term alone first.
Divide both sides by $-2$.
$y = 3x - 5$
Now both equations are in the form $y = mx + b$.
Compare the slopes and the intercepts.
State the answer.
The graphs of $y = 4x + 4$ and $y = -2x + 28$ cross at $(4, 20)$. What does that point tell you?
Complete the worked solution of the system $x + y = 75$ and $y = 4x$.
Replace $y$ in the first equation with $4x$.
$x + 4x = 75$
The second equation says $y$ is the same as $4x$.
Collect the $x$ terms.
p $x = 75$
One $x$ plus $4$ more $x$ is a single $x$ term.
Divide both sides by the coefficient of $x$.
$x =$ q
Dividing undoes the multiplying.
Use $y = 4x$ to find $y$.
$y =$ r
Multiply the value of $x$ by $4$; the two values should add to $75$.
Solve the system $y = 5x + 4$ and $y = -2x + 25$.
$x =$ x, $y =$ y
Two numbers add to $43$ and differ by $9$. What are they?
The larger number is a and the smaller is b.
How many solutions does the system $y = 5x + 7$ and $4y = 20x + 8$ have?
Solve the system $2x + 4y = -18$ and $x - 4y = 21$.
$x =$ x, $y =$ y
A middle school sold $88$ tickets for its spring musical and took in $464$ dollars. Adult tickets cost $13$ dollars and student tickets cost $3$ dollars. How many of each were sold?
a adult tickets and s student tickets
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
Solve the system $y = 2x - 3$ and $5x + 3y = -20$.
$x =$ x, $y =$ y
You can solve a system of two linear equations. Without looking: what does the solution mean on a graph, and what does it mean if the two lines are parallel?
18. Your turn: how many solutions do $y = 3x + 2$ and $6x - 2y = 10$ have?, step 3
$\text{slopes } 3, 3; \quad \text{intercepts } 2, -5$
Same slope, different intercepts: the lines are parallel.
18. Your turn: how many solutions do $y = 3x + 2$ and $6x - 2y = 10$ have?, step 4
$\text{no solution}$
Parallel lines never cross, so no pair satisfies both.