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Substitute Values into an Equation and Solve for a Variable
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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.

Substitute Values into an Equation and Solve for a Variable

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

### Practice Problems

1. A train traveled for $$t=5$$ hours at a constant speed of $$r=60$$ miles per hour. Use the formula $$d=r \cdot t$$ to find the total distance ($$d$$) the train traveled (in miles). (
Solution
Solution: $$d = r \cdot t = 60 \cdot 5 = \mathbf{300 \text{ miles}}$$
Details:
We start by plugging the numbers we’ve been given into our formula for distance.

Plug 60 miles per hour in for our rate $$r$$.

Plug 5 hours in for our time $$t$$.

$$d = (r)(t)$$

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.

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.

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

$$(60)(5)= 300$$

Our final solution: 300 miles.
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2. A man walked a distance of $$d=15km$$ (kilometers) in $$t=3$$ hours at a constant rate. Use the formula $$d=r \cdot t$$ to find the speed ($$r$$) of the man in km per hour. (
Solution
Solution:
$$r = d \div t = 15 \div 3 = \mathbf{5\;km\text{ per hour}}$$
)
3. A boat traveled a distance of $$d=140$$ miles at a constant speed of $$r=70$$ miles per hour. Use the formula $$d=r \cdot t$$ to find the number of hours ($$t$$) that the trip took. (
Solution
Solution: $$t = d \div r = 140 \div 70 = \mathbf{2 \text{ hours}}$$
Details:
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.

$$d = r\cdot t$$

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

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.

Next, we simplify both sides of the equation.

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

Our final solution: $$t = 2\text{ hours}$$.
)
4. A bus traveled for $$t=3.1$$ hours at a constant speed of $$r=62$$ miles per hour. Use the formula $$d=r \cdot t$$ to find the total distance ($$d$$) the bus traveled (in miles). Round your answer to the nearest tenth. (
Video Solution
Solution: $${\text{d}} = {\text{r}} \cdot {\text{t}} = 62 \cdot 3.1 =$$ 192.2 miles
Details:

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5. A truck traveled a distance of $$d=615km$$ (kilometers) over $$t=5.9$$ hours at a constant rate. Use the formula $$d=r \cdot t$$ to find the speed ($$r$$) of the truck in km per hour. Round your answer to the nearest tenth. (
Solution
Solution: $$r = d \div t = 615 \div 5.9 = \mathbf{104.2\;km \text{ per hour}}$$
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6. A bicyclist traveled a distance of $$d=18.6$$ miles at a constant speed of $$r=16$$ miles per hour. Use the formula $$d=r \cdot t$$ to find the number of hours ($$t$$) that the trip took. Round your answer to the nearest tenth. (
Video Solution
Solution: $${\text{t}} = {\text{d}} \div {\text{r}} = 18.6 \div 16 =$$ 1.2 hours
Details:

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## Need More Help?

1. Study other Math Lessons in the Resource Center.
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