Linear equations in two variables

There's one equation with two letters, usually $x$ and $y$, and its graph is a straight line in the coordinate planeThe grid a graph is drawn on, with one axis across and one up.. If the question names a functionA rule that takes in one number and gives back one number, like f(x) = 5x - 7.Read the lesson like $f(x)$, you want Linear functions. If it gives two equations to solve together, that's a system of equationsTwo or more equations that all have to be true at the same time.Read the lesson, covered in Systems of two linear equations.

Read the last line first. It might say:

• “Which equation defines line $k$?”
• “What is the slope of line $j$?”
• “What is the $y$-coordinate of the $y$-intercept of the graph of the given equation?”
• “Which table gives three values of $x$ and their corresponding values of $y$ for the given equation?”
• “Which equation represents this situation?”
• “Which of the following is the best interpretation of 9 in this context?”

A point is an ordered pairTwo numbers that give a point’s position, like (2, 70) for 2 across and 70 up. $(x, y)$, with $x$ first. A row of a table is a point, and so is a labeled dot on a graph.

In $y=mx+b$, $m$ is the slope and the line crosses the vertical axis at $(0,b)$.

Many equations come as $ax+by=c$ instead. Get $y$ alone, then read the slope and interceptWhere a graph crosses an axis.Read the lesson.

1. Start.

   $$15x+3y=24$$

2. Subtract $15x$ from both sides.

   $$3y=-15x+24$$

3. Divide every term by 3.

   $$y=-5x+8$$

Multiplying every term by the same number gives the same line. $15x+3y=24$, $5x+y=8$ and $30x+6y=48$ all have slope $-5$ and cross at $(0,8)$. If no choice looks like your answer, divide each one through by a number every term shares, or compare their $-\frac{a}{b}$.

An intercept is where the line crosses an axis. To find one, set the other letter to 0. At the y-intercept $x=0$, and at the x-interceptThe value of x where a graph crosses the horizontal axis, so the output there is 0. $y=0$.

Putting in the 0 first saves the most work when each letter has its own fraction. Say a question asks for the $y$-coordinate of the y-intercept of $\frac{3y}{4}-\frac{5x}{6}=9$.

1. Put in $x=0$.

   $$\frac{3y}{4}-\frac{5(0)}{6}=9$$

2. The whole $x$ term is 0, including the 6.

   $$\frac{3y}{4}=9$$

3. Multiply both sides by 4.

   $$3y=36$$

4. Divide both sides by 3.

   $$y=12$$

For an x-intercept, type y=0 instead. The point shows decimals, so $\frac85$ reads as 1.6, which is fine on a grid-inA question where you type your own answer instead of picking one of four.Read the lesson.

Sometimes the equation has another letter in it, and you're given a point on the line. Put the point in first to find the letter, then answer the question.

Say $4x+ay=30$ passes through $(3,6)$, and the question asks for the slope.

1. Put in $x=3$ and $y=6$.

   $$4(3)+a(6)=30$$

2. Multiply.

   $$12+6a=30$$

3. Subtract 12 from both sides.

   $$6a=18$$

4. Divide both sides by 6.

   $$a=3$$

5. Put $a=3$ back in.

   $$4x+3y=30$$

6. Subtract $4x$ from both sides.

   $$3y=-4x+30$$

7. Divide every term by 3. The slope is $-\frac43$.

   $$y=-\frac43x+10$$

A line passes through $(-5,-8)$ and $(5,-2)$. Which equation describes it?

• A. $y=\frac53x-5$
• B. $y=-5x+\frac35$
• C. $y=\frac35x-11$
• D. $y=\frac35x-5$

1. Slope: change in $y$ over change in $x$, subtracting in the same order on top and bottom.

   $$m=\frac{-2-(-8)}{5-(-5)}$$

2. Simplify.

   $$m=\frac{6}{10}=\frac35$$

3. Put $(-5,-8)$ into $y=mx+b$.

   $$-8=\frac35(-5)+b$$

4. $\frac35$ times $-5$ is $-3$.

   $$-8=-3+b$$

5. Add 3 to both sides.

   $$b=-5$$

6. Check the other point. It gives $-2$, so it fits.

   $$\frac35(5)-5=-2$$

D is correct. Don't use $-8$ as $b$, because only a point with $x=0$ gives you $b$ directly. Linear functions walks through this same build with a story.

B swaps the slope and the intercept. C comes from the other way to finish a line, point-slope form, $y-y_1=m(x-x_1)$.

1. Put in $m=\frac35$ and the point $(-5,-8)$.

   $$y-(-8)=\frac35\bigl(x-(-5)\bigr)$$

2. Subtracting a negative is adding.

   $$y+8=\frac35(x+5)$$

3. Multiply both terms by $\frac35$. $\frac35$ times 5 is 3.

   $$y+8=\frac35x+3$$

4. Subtract 8 from both sides.

   $$y=\frac35x-5$$

Writing $-3$ in the third row gives choice C. Use whichever route you make fewer sign slips in.

To test the choices in Desmos, type the two points and graph each choice. Any line that misses a dot is out. Linear functions shows the keys. Try one built from a graph.

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At Horvath’s scrapyard, the graph relates the number $x$ of identical steel couplings in a collection bin to the total weight $y$, in kilograms, of the bin and couplings. The empty bin has a fixed weight. Two recorded points on the line are $(3, 12)$ and $(6, 17)$.

Which equation represents this situation?

1. A

   $y = \frac{5}{3}x + 7$

2. B

   $y = \frac{3}{5}x + \frac{51}{5}$

3. C

   $y = 7x + \frac{5}{3}$

4. D

   $y = \frac{5}{3}x + 17$

Show the answer

• A

  $y = \frac{5}{3}x + 7$

  Correct answer

  The slopeHow steep a line is, measured by how much it goes up for each unit it moves to the right.Read the lesson is $\frac{17-12}{6-3}=\frac{5}{3}$. Using $(3,12)$, the interceptWhere a graph crosses an axis.Read the lesson is $12-\frac{5}{3}(3)=7$, so the equationTwo expressions set equal to each other, like 5x + 7 = 22.Read the lesson is $y=\frac{5}{3}x+7$.

• B

  $y = \frac{3}{5}x + \frac{51}{5}$

  This equationTwo expressions set equal to each other, like 5x + 7 = 22.Read the lesson reverses the slopeHow steep a line is, measured by how much it goes up for each unit it moves to the right.Read the lesson calculation, using $\frac{\Delta x}{\Delta y}=\frac{3}{5}$ instead of $\frac{\Delta y}{\Delta x}=\frac{5}{3}$. The interceptWhere a graph crosses an axis.Read the lesson $\frac{51}{5}$ then comes from using that reciprocal slope with $(3,12)$.

• C

  $y = 7x + \frac{5}{3}$

  This equationTwo expressions set equal to each other, like 5x + 7 = 22.Read the lesson swaps the roles of the slopeHow steep a line is, measured by how much it goes up for each unit it moves to the right.Read the lesson and interceptWhere a graph crosses an axis.Read the lesson. It places the intercept $7$ as the coefficientThe number multiplying a letter, like the 5 in 5x.Read the lesson of $x$ and the slope $\frac{5}{3}$ as the constantA number that doesn’t change, like the 7 in 5x + 7, or a letter such as p that stands for one fixed value.Read the lesson term.

• D

  $y = \frac{5}{3}x + 17$

  This equationTwo expressions set equal to each other, like 5x + 7 = 22.Read the lesson has a sign error when $\frac{5}{3}(x-3)$ is distributed. The correction term should be $-5$, but changing it to $+5$ produces the incorrect interceptWhere a graph crosses an axis.Read the lesson $17$.

Solution

Use the two recorded points to calculate the slopeHow steep a line is, measured by how much it goes up for each unit it moves to the right.Read the lesson:

$m=\frac{17-12}{6-3}=\frac{5}{3}$.

Substitute $m=\frac{5}{3}$ and $(3,12)$ into $y=mx+b$:

$12=\frac{5}{3}(3)+b$

$12=5+b$

$b=7$.

The $y$-intercept also represents the weight of the empty bin, so an interceptWhere a graph crosses an axis.Read the lesson of $7$ agrees with the graph. The equationTwo expressions set equal to each other, like 5x + 7 = 22.Read the lesson is $y=\frac{5}{3}x+7$.

Report a problem

A table gives you points.

The $y$-values can have a letter in them. With rows $(1,n+6)$ and $(4,n+24)$, the $n$ cancels.

1. Change in $y$ over change in $x$.

   $$m=\frac{(n+24)-(n+6)}{4-1}$$

2. $n-n=0$ and $24-6=18$.

   $$m=\frac{18}{3}$$

3. Divide.

   $$m=6$$

When the choices are tables, put each row's $x$ into the given equation and see if you get its $y$. Here's a choice for $y=-2x+9$.

A print shop starts the week with 15 boxes of paper. Each order it prints uses 3 reams, and a box holds 5 reams. Say a graph shows the boxes left, $y$, after $x$ orders, with the labeled points $(0,15)$ and $(10,9)$.

The slope from the points is $\frac{9-15}{10-0}=-\frac{6}{10}=-\frac35$. The story gives the same thing: 3 reams is $\frac35$ of a box, and the paper is being used up, so the slope is $-\frac35$. The line is $y=-\frac35x+15$.

The point $(a,3)$ is on the line. What is the value of $a$?

1. Put in $y=3$, with $a$ for $x$.

   $$3=-\frac35a+15$$

2. Subtract 15 from both sides.

   $$-12=-\frac35a$$

3. Multiply both sides by $-\frac53$.

   $$a=20$$

Some questions add a step after the line. If one asks how many reams are left after 5 orders, find the boxes first: $-\frac35(5)+15=12$. Then change units: 12 boxes of 5 reams is 60 reams. If a story has two machines or two workers, write each one's rate next to its name before you compare them.

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Aslan picks 2 mushroom crates per minute. The graph records crates remaining after each 4-minute period.

The point with coordinates $(19,y)$ lies on the line shown. What is the value of $y$?

Show the answer

Accepted answer: 72

Solution

The slopeHow steep a line is, measured by how much it goes up for each unit it moves to the right.Read the lesson is $\frac{176-224}{6-0}=\frac{-48}{6}=-8$, so the number of crates decreases by $8$ during each 4-minute period. Since the line passes through $(0,224)$, its equationTwo expressions set equal to each other, like 5x + 7 = 22.Read the lesson is $y=224-8x$.

Substitute $x=19$: $y=224-8(19)=224-152=72$.

A common wrong answer is $186$, which uses the rate of $2$ crates per minute as the slope magnitude without converting it to $8$ crates per 4-minute period. Another common wrong answer is $19$, which reports the given $x$-coordinate instead of calculating the requested $y$-coordinate.

Report a problem

Some questions give the line as $ax+by=c$ and write the choices that way too.

Which line is perpendicular to $4x+5y=10$ and passes through $(2,6)$?

• A. $5x+4y=34$
• B. $4x-5y=-22$
• C. $5x-4y=-8$
• D. $10x-8y=-28$

1. The slope of $4x+5y=10$ is $-\frac{a}{b}$.

   $$m=-\frac45$$

2. Flip it and change the sign.

   $$m=\frac54$$

3. Put in $(2,6)$.

   $$6=\frac54(2)+b$$

4. Multiply.

   $$6=\frac52+b$$

5. Subtract $\frac52$. 6 is $\frac{12}{2}$.

   $$b=\frac72$$

6. Multiply every term of $y=\frac54x+\frac72$ by 4.

   $$4y=5x+14$$

7. Subtract $5x$ from both sides, then multiply every term by $-1$.

   $$5x-4y=-14$$

8. Double every term. That's D.

   $$10x-8y=-28$$

D passes through $(2,6)$, since $10(2)-8(6)=-28$. A flips the slope without changing its sign, and B changes the sign without flipping. C has the right slope, but it keeps the original line's intercept, 2, when it should pass through $(2,6)$.

Two equations in this form give a faster test. $a_1x+b_1y=c_1$ and $a_2x+b_2y=c_2$ are perpendicular when $a_1a_2+b_1b_2=0$: for $4x+5y=10$ and $5x-4y=-14$, $4(5)+5(-4)=0$. They're parallel when $a_1b_2=a_2b_1$. The test helps most when a letter sits in a coefficientThe number multiplying a letter, like the 5 in 5x.Read the lesson. Say $4x+ay=9$ and $(a+5)x+3y=2$ are perpendicular.

1. Multiply the $x$ numbers, multiply the $y$ numbers, and add.

   $$4(a+5)+a(3)=0$$

2. Multiply out $4(a+5)$.

   $$4a+20+3a=0$$

3. Combine like terms, then subtract 20 from both sides.

   $$7a=-20$$

4. Divide both sides by 7.

   $$a=-\frac{20}{7}$$

Try one with the choices in this form.

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A streaming service models basic subscribers $x$ and premium subscribers $y$, in thousands, using line $r$. This line contains the table’s point and is perpendicular to $3x + 4y = 180$.

Which equation defines line $r$?

1. A

   $8x + 6y = 570$

2. B

   $8x - 6y = -90$

3. C

   $6x - 8y = -260$

4. D

   $8x - 6y = -270$

Show the answer

• A

  $8x + 6y = 570$

  You took the reciprocal of $-\frac{3}{4}$ to get $-\frac{4}{3}$ without changing its sign. This equationTwo expressions set equal to each other, like 5x + 7 = 22.Read the lesson passes through $(30,55)$, but perpendicular slopesHow steep a line is, measured by how much it goes up for each unit it moves to the right.Read the lesson must have a product of $-1$.

• B

  $8x - 6y = -90$

  Correct answer

  The given line has slopeHow steep a line is, measured by how much it goes up for each unit it moves to the right.Read the lesson $-\frac{3}{4}$, so a perpendicular line has slope $\frac{4}{3}$. Using $(30,55)$ gives $55=\frac{4}{3}(30)+b$, so $b=15$. Thus $y=\frac{4}{3}x+15$, which rearranges to $8x-6y=-90$.

• C

  $6x - 8y = -260$

  You changed the original slopeHow steep a line is, measured by how much it goes up for each unit it moves to the right.Read the lesson’s sign from $-\frac{3}{4}$ to $\frac{3}{4}$ without reversing the fraction. This equationTwo expressions set equal to each other, like 5x + 7 = 22.Read the lesson fits $(30,55)$, but the perpendicular slope requires both changes.

• D

  $8x - 6y = -270$

  You used the perpendicular slopeHow steep a line is, measured by how much it goes up for each unit it moves to the right.Read the lesson $\frac{4}{3}$ but kept the original line’s interceptWhere a graph crosses an axis.Read the lesson of $45$. That gives $y=\frac{4}{3}x+45$, which yields $y=85$ when $x=30$, missing the table’s point.

Solution

Solve the given equationTwo expressions set equal to each other, like 5x + 7 = 22.Read the lesson for $y$: $3x+4y=180$ gives $y=-\frac{3}{4}x+45$. Its slopeHow steep a line is, measured by how much it goes up for each unit it moves to the right.Read the lesson is $-\frac{3}{4}$, so the slope of perpendicular line $r$ is the negative reciprocal, $\frac{4}{3}$.

The table gives the point $(30,55)$. Substitute it into $y=\frac{4}{3}x+b$: $55=\frac{4}{3}(30)+b=40+b$, so $b=15$.

Therefore, $r$ has equation $y=\frac{4}{3}x+15$. Multiply by $6$ to get $6y=8x+90$, then rearrange to obtain $8x-6y=-90$.

Report a problem

Moving a line up or down changes where it crosses the vertical axis and leaves the slope alone. Say the graph of $14x+21y=126$ is shifted down 4 units, and the question asks for the new y-intercept.

1. Put in $x=0$ to find the original y-intercept.

   $$21y=126$$

2. Divide both sides by 21.

   $$y=6$$

3. Down 4 takes 4 off. The new y-intercept is $(0,2)$.

   $$6-4=2$$

If the question asks for the new x-intercept, write the shifted line and set $y=0$.

1. Solve the original for $y$: $-\frac{14}{21}=-\frac23$ and $\frac{126}{21}=6$.

   $$y=-\frac23x+6$$

2. Down 4: take 4 off the intercept.

   $$y=-\frac23x+2$$

3. Set $y=0$.

   $$0=-\frac23x+2$$

4. Add $\frac23x$ to both sides, then multiply both sides by $\frac32$.

   $$x=3$$

To move a line right $h$ units, replace $x$ with $x-h$. Moving $y=2x+3$ right 4 gives $y=2(x-4)+3$, which is $y=2x-5$. Moving left $h$ uses $x+h$.

Each loaf a baker shapes takes 7 minutes and each tray of rolls takes 5 minutes. She shapes $b$ loaves and $r$ trays, 258 minutes in all, so $7b+5r=258$.

If you're given one count, put it in and solve for the other. Say she shaped 18 trays.

1. Put in $r=18$.

   $$7b+5(18)=258$$

2. Multiply.

   $$7b+90=258$$

3. Subtract 90 from both sides.

   $$7b=168$$

4. Divide both sides by 7.

   $$b=24$$

Sometimes each rate has a constantA number that doesn’t change, like the 7 in 5x + 7, or a letter such as p that stands for one fixed value.Read the lesson in it, like $(p+9)a+(p+4)k=350$ for the cost of adult and kids' tickets. If the question asks how much more an adult ticket costs, subtract: $(p+9)-(p+4)=5$. The $p$ cancels, so you never need to find it.

Before each one, decide whether you're reading the equation, building it, relating two lines, or saying what a number means. When you build an equation, check it against a point you didn't use.