Solving for a Variable:

Substitute Values into an Equation and Solve for a Variable

Sometimes we are given an equation with multiple variables, as in multiple letters. Most of these variables will be known, so we can replace the variables in our equation with the numbers that we know they are equal to. Once we’ve replaced these variables with numbers, we can solve for whichever variable is left. The following video will show you how to do this with a few examples.

Video Source (07:40 mins) | Transcript

Once we substitute the values we know into the equation, solving the equation is just the same as what we’ve learned in the lessons before this.

Additional Resources

Khan Academy: Finding Average Speed or Rate (13:14 mins, Transcript)

Practice Problems

A train traveled for $\boldsymbol{\text{t}}=5$ hours at a constant speed of $\boldsymbol{\text{r}}=60$ miles per hour. Use the formula $\boldsymbol{\text{d}} = \boldsymbol{\text{r}} \cdot \boldsymbol{\text{t}}$ to find the total distance ($\boldsymbol{\text{d}}$) the train traveled (in miles).

A man walked a distance of $\boldsymbol{\text{d}}=15{\text{km}}$ (kilometers) in $\boldsymbol{\text{t}}=3$ hours at a constant rate. Use the formula $\boldsymbol{\text{d}} = \boldsymbol{\text{r}} \cdot \boldsymbol{\text{t}}$ to find the speed ($\boldsymbol{\text{r}}$ of the man in km per hour.

A boat traveled a distance of $\boldsymbol{\text{d}}=140$ miles at a constant speed of $\boldsymbol{\text{r}}=70$ miles per hour. Use the formula $\boldsymbol{\text{d}} = \boldsymbol{\text{r}} \cdot \boldsymbol{\text{t}}$ to find the number of hours ($\boldsymbol{\text{t}}$) that the trip took.

A bus traveled for $\boldsymbol{\text{t}}=3.1$ hours at a constant speed of $\boldsymbol{\text{r}}=62$ miles per hour. Use the formula $\boldsymbol{\text{d}} = \boldsymbol{\text{r}} \cdot \boldsymbol{\text{t}}$ to find the total distance ($\boldsymbol{\text{d}}$) the bus traveled (in miles). Round your answer to the nearest tenth.

A truck traveled a distance of $\boldsymbol{\text{d}}=615{\text{km}}$ (kilometers) over $\boldsymbol{\text{t}}=5.9$ hours at a constant rate. Use the formula $\boldsymbol{\text{d}} = \boldsymbol{\text{r}} \cdot \boldsymbol{\text{t}}$ to find the speed ($\boldsymbol{\text{r}}$) of the truck in km per hour. Round your answer to the nearest tenth.

A bicyclist traveled a distance of $\boldsymbol{\text{d}}=18.6$ miles at a constant speed of $\boldsymbol{\text{r}}=16$ miles per hour. Use the formula $\boldsymbol{\text{d}} = \boldsymbol{\text{r}} \cdot \boldsymbol{\text{t}}$ to find the number of hours ($\boldsymbol{\text{t}}$) that the trip took. Round your answer to the nearest tenth.

Solutions

${\text{d}} = {\text{r}} \cdot {\text{t}} = 60 \cdot 5 =$ 300 miles (Written Solution)

We start by plugging the numbers we’ve been given into our formula for distance.

Plug 60 miles per hour in for our rate ${\text{r}}$.

Plug 5 hours in for our time ${\text{t}}$.

${\text{d}} = ({\text{r}})({\text{t}})$

$The image has the amounts for the variables and their units inserted into the parentheses. The amounts and units are displayed as fractions within parentheses. It says d equals (60 miles over one hour)(five hours over one). The parentheses next to each other indicate multiplication between the fractions.$

As in unit conversions, our units on the top of the fraction and units on the bottom of the fraction that are the same can cancel out.

$This image shows the equation of variable d equals (60 miles over one hour)(five hours over one). The parentheses next to each other indicate multiplication between the fractions. The units of hours on the bottom of the first fraction and the units of hours on the top of the second fraction are both crossed out with a red line.$

Now we multiply across. Since the denominators are both 1, and anything divided by 1 is still itself, we just need to multiply 60 times 5.

$This image shows the equation of variable d equals (60 miles over one hour)(five hours over one). The parentheses next to each other indicate multiplication between the fractions. The units of hours on the bottom of the first fraction and the units of hours on the top of the second fraction are both crossed out with a red line.$

${\text{d}} = {\color{Cyan} (60 \text{ miles})(5)}$

$(60)(5)= 300$

Our final solution: 300 miles

${\text{r}} = {\text{d}} \div {\text{t}} = 15 \div 3 =$ 5 km per hour

${\text{t}} = {\text{d}} \div {\text{r}} = 140 \div 70 =$ 2 hours (Written Solution)

In this example, we are given the distance and the rate of speed, but we need to find the time.

First, substitute in the numbers that we know.

${\text{d}} = {\text{r}}\cdot {\text{t}}$

$140 {\text{ miles}} = 70{\text{ miles per hour}}\cdot ({\text{t}}) $

$This image has two equations written one on top of the other. The top equation is variable d equals variable r dot variable t equals (r)(t). Below this equation is the rewritten version with the given values inserted in for variables d and r. It is 140 miles equals (seventy miles over one hour)(t). The seventy miles over one hour is written as a fraction and in the color blue to indicate it is new. The 140 miles is also written in blue.$

Next, solve for the variable t.

In order to get the variable t all by itself on the right-hand side of the equation, we can multiply by the multiplicative inverse of 70 miles per hour which is 1hour per 70 miles.

$This equation is the same as the previous one except the fraction (one hour over seventy miles) is multiplied to both sides of the equation and written in blue on both sides to indicate it is new. Written out it is (one hour over seventy miles) times 140 miles equals (one hour over seventy miles) times (seventy miles over one hour) times (t).$

Next, we simplify both sides of the equation.

On the right-hand side, the multiplicative inverses become 1 leaving us with just the variable t.

$This image shows three equations written one over the other. The top equation is the same as the equation in the previous image. Written out it is (one hour over seventy miles) times 140 miles equals (one hour over seventy miles) times (seventy miles over one hour) times (t). Under the right-hand side, just under the portion of the equation that is the multiplication of the two fractions (one hour over seventy miles) times (seventy miles over one hour), there is a green bracket with an arrow pointing down to the right-hand side of the next equation. The right-hand side of this next equation is one times t. The one is in green showing it is the result of multiplying the two previous fractions together. To the left of this is the equal sign. It is equal to the left-hand side of the equation which is written as a fraction with (one hour)(140 miles) in the numerator and 70 miles in the denominator. At the very bottom of the image is the final equation which is two hours equals t.$

Our final solution: ${\text{t}}=2$ hours

${\text{d}} = {\text{r}} \cdot {\text{t}} = 62 \cdot 3.1 =$ 192.2 miles (Solution Video | Transcript)

${\text{r}} = {\text{d}} \div {\text{t}} = 615 \div 5.9 = $ 104.2 km per hour

${\text{t}} = {\text{d}} \div {\text{r}} = 18.6 \div 16 =$ 1.2 hours (Solution Video | Transcript)