Writing Equations for Proportional Relationships | Grade 7 Math | LevelKite

A table lists only a few pairs of values, and a graph labels only a few points. An equation is a math sentence that says two amounts are equal. One equation can describe every pair of a proportional relationship at once. In this lesson you will write that equation, first from a table and then from a graph.

You already have the main tool. Two quantities $x$ and $y$ are in a proportional relationship when the ratio $\frac{y}{x}$ is the same number for every pair of values with $x \neq 0$ , and $y$ is $0$ when $x$ is $0$ . That number is the constant of proportionality, written $k$ . You have found $k$ from tables and from graphs before. Now you will use it to build an equation.

Start with the table that follows. It shows the cost of notebooks at a school store, where every notebook has the same price.

Notebooks bought and their total cost

Number of notebooks, x	Total cost in dollars, y	Ratio of y to x	y written as 3 times x
$2$	$6$	$\frac{6}{2} = 3$	$6 = 3 \cdot 2$
$5$	$15$	$\frac{15}{5} = 3$	$15 = 3 \cdot 5$
$8$	$24$	$\frac{24}{8} = 3$	$24 = 3 \cdot 8$

In the table, $x$ is the number of notebooks and $y$ is the total cost in dollars. The third column shows that $\frac{y}{x} = 3$ in every row, so $k = 3$ . This is the unit rate, the amount of $y$ for $1$ unit of $x$ : each notebook costs $3$ dollars.

Now read the last column. Dividing $y$ by $x$ gives $3$ , so multiplying $x$ by $3$ gives $y$ back. One sentence is true in every row: the total cost equals 3 times the number of notebooks. The letters $x$ and $y$ are variables, which means they stand for numbers that can change. Written with the variables, that sentence is the equation $y = 3x$ .

Here is another way to see it. No $x$ in the table is $0$ , so $\frac{y}{x} = 3$ makes sense in every row. Multiplying both sides by the same number, $x$ , keeps them equal and gives $y = 3x$ . The equation even covers $0$ notebooks, a pair the ratio cannot test because you cannot divide by $0$ : $y = 3 \cdot 0 = 0$ dollars.

Key concept

The equation of a proportional relationship

In a proportional relationship between $x$ and $y$ , the ratio $\frac{y}{x}$ equals one constant $k$ for every pair with $x \neq 0$ . Multiplying both sides of $\frac{y}{x} = k$ by $x$ gives the equation $y = kx$ , which also fits the pair $(0, 0)$ because $k \cdot 0 = 0$ .

On a graph, $y = kx$ is a straight line through the origin, the point $(0, 0)$ . The point $(1, k)$ lies on that line and shows the unit rate.

To write the equation, find $k = \frac{y}{x}$ from one pair with $x \neq 0$ , check that every other pair with $x \neq 0$ gives the same $k$ , and then replace $k$ with that number.

y = kx

Filling a pool with a hose

Time in minutes, x	Water in gallons, y	Ratio of y to x
$3$	$24$	$\frac{24}{3} = 8$
$5$	$40$	$\frac{40}{5} = 8$
$10$	$80$	$\frac{80}{10} = 8$

Worked example

From a table to an equation

A hose fills an empty pool at a steady rate, so the amount of water is proportional to the time. The table just before this example shows the time $x$ , in minutes, and the water $y$ , in gallons. Write an equation for this relationship.

Say what the variables mean: x is the time in minutes and y is the water in gallons. The relationship is proportional, so its equation has the form y = kx, where k is the constant of proportionality.
$y = kx$
Find k from the first row by dividing y by x. You can divide because x = 3 is not 0.
$k = \frac{24}{3} = 8$
Check the other rows, because k must be the same for every pair where x is not 0. The third column of the table shows the work: both ratios are also 8.
$\frac{40}{5} = 8 \qquad \frac{80}{10} = 8$
Replace k with 8 in y = kx. The 8 is the unit rate: the hose adds 8 gallons in each 1 minute.
$y = 8x$
Test the equation with a row from the table. For x = 10, multiply 8 by 10. The result is 80, which matches the 80 gallons in the table, so the equation fits.
$y = 8 \cdot 10 = 80$

Answer

The equation is $y = 8x$ , where $x$ is the time in minutes and $y$ is the water in gallons. The hose adds $8$ gallons each minute.

A graph shows the same kind of information as a table. Each point $(x, y)$ is one pair of values, just like one row. When the relationship is proportional, $y = kx$ , so the points lie on a straight line through the origin $(0, 0)$ . The point with $x = 1$ is $(1, k)$ , and it shows the unit rate.

In the graph that follows, a toy robot rolls across the floor at a steady speed. The horizontal axis shows the time $x$ , in seconds. The vertical axis shows the distance $y$ , in feet. Time and distance cannot be negative here, so the graph shows only values of $0$ or more. Line A is straight, and it passes through the origin and three more labeled points.

Treat each labeled point like a row of a table. Use the exact numbers in its label, never a guess from the grid.

Line A passes through

(0, 0)

,

(1, 2)

,

(3, 6)

and

(5, 10)

. Time in seconds runs across and distance in feet runs up.

Worked example

From a graph to an equation

Line A shows the robot's trip: $x$ is the time in seconds and $y$ is the distance in feet. Use the labeled points $(0, 0)$ , $(1, 2)$ , $(3, 6)$ and $(5, 10)$ to write an equation for the relationship.

Check the two features that the graph of every proportional relationship has. Line A is straight, and it passes through the labeled origin (0, 0), so 0 seconds goes with 0 feet. Line A has both features, so look for the constant ratio next.
Divide y by x at each labeled point where x is not 0, because in a proportional relationship that ratio is k. Use the exact numbers in the labels.
$\frac{2}{1} = 2 \qquad \frac{6}{3} = 2 \qquad \frac{10}{5} = 2$
All three ratios equal 2, so k = 2. The labeled point (1, 2) shows this directly: it is the point (1, k), and it tells you the robot rolls 2 feet in 1 second, the unit rate.
$k = 2$
Replace k with 2 in y = kx, the form of every proportional relationship.
$y = 2x$
Test the equation with the origin and with (5, 10). The ratio cannot test the origin, because you cannot divide by 0, but the equation can. Both points fit.
$2 \cdot 0 = 0 \qquad 2 \cdot 5 = 10$

Answer

The equation is $y = 2x$ , where $x$ is the time in seconds and $y$ is the distance in feet. The robot rolls $2$ feet each second.

Sometimes no labeled point has $x = 1$ , so you cannot read $k$ straight from a label. Dividing still works, because in a proportional relationship $\frac{y}{x}$ is the same at every point with $x \neq 0$ .

In the next graph, the horizontal axis shows the weight $x$ , in pounds, of apples you buy. The vertical axis shows the cost $y$ , in dollars. Weight and cost cannot be negative, so again the graph shows only values of $0$ or more. Line B is straight, and it passes through the origin and three more labeled points.

Line B passes through

(0, 0)

,

(2, 3)

,

(4, 6)

and

(6, 9)

. Weight in pounds runs across and cost in dollars runs up.

Worked example

When no labeled point has x = 1

Line B shows the cost of apples: $x$ is the weight in pounds and $y$ is the cost in dollars. Use the labeled points $(0, 0)$ , $(2, 3)$ , $(4, 6)$ and $(6, 9)$ to write an equation for the relationship. Then find the unit rate.

Line B is straight and passes through the labeled origin (0, 0), so it has both features of the graph of a proportional relationship. Divide y by x at each labeled point where x is not 0 to find k.
$\frac{3}{2} = 1.5 \qquad \frac{6}{4} = 1.5 \qquad \frac{9}{6} = 1.5$
All three ratios equal 1.5, so k = 1.5. Replace k with 1.5 in y = kx.
$y = 1.5x$
No labeled point has x = 1, so compute the unit rate instead of guessing from the grid. Use x = 1 in the equation. The result is the pair (1, 1.5), which is the point (1, k): 1 pound costs 1.5 dollars.
$y = 1.5 \cdot 1 = 1.5$
Test the equation with the origin and with (6, 9). Both points fit.
$1.5 \cdot 0 = 0 \qquad 1.5 \cdot 6 = 9$

Answer

The equation is $y = 1.5x$ , where $x$ is the weight in pounds and $y$ is the cost in dollars. The unit rate is $1.5$ dollars for each pound, which is 1 dollar and 50 cents.

The big idea: in a proportional relationship, $\frac{y}{x}$ equals one constant $k$ for every pair with $x \neq 0$ . That single number gives you the whole equation, $y = kx$ .

From a table, divide $y$ by $x$ in one row with $x \neq 0$ , then check the other rows.
From a graph, check that the line is straight and passes through $(0, 0)$ . Then divide $y$ by $x$ at labeled points, or read $k$ from the point $(1, k)$ .
Test your equation with a pair you know.

Next, you will interpret what the points on the graph of a proportional relationship mean, and you will graph proportional relationships from word problems.