Solve a Formula for a Specific Variable

2.3 Solve a Formula for a Specific Variable

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2.3.1 Solve a Formula for a Specific Variable

We have all probably worked with some geometric formulas in our study of mathematics. Formulas are used in so many fields, it is important to recognize formulas and be able to manipulate them easily.

It is often helpful to solve a formula for a specific variable. If you need to put a formula in a spreadsheet, it is not unusual to have to solve it for a specific variable first. We isolate that variable on one side of the equals sign with a coefficient of one and all other variables and constants are on the other side of the equal sign.

Geometric formulas often need to be solved for another variable, too. The formula $V=\frac{1}{3}πr^{2}h$ is used to find the volume of a right circular cone when given the radius of the base and height. In the next example, we will solve this formula for the height.

Example 1

Solve the formula $V=\frac{1}{3}πr^{2}h$ for $h$.

Solution

Write the formula.$V=\frac{1}{3}πr^{2}h$
Remove the fraction on the right.$3 \cdot V=3 \cdot \frac{1}{3}πr^{2}h$
Simplify.$3V=πr^{2}h$
Divide both sides by $πr^{2}$.$\frac{3V}{πr^{2}}=h$

We could now use this formula to find the height of a right circular cone when we know the volume and the radius of the base, by using the formula $h=\frac{3V}{πr^{2}}$.

In the sciences, we often need to change temperature from Fahrenheit to Celsius or vice versa. If you travel in a foreign country, you may want to change the Celsius temperature to the more familiar Fahrenheit temperature.

Example 2

Solve the formula $C=\frac{5}{9}(F-32)$ for $F$.

Solution

Write the formula.$C=\frac{5}{9}(F-32)$
Remove the fraction on the right.$\frac{9}{5}C=\frac{9}{5} \cdot \frac{5}{9}(F-32)$
Simplify.$\frac{9}{5}C=(F-32)$
Add $32$ to both sides.$\frac{9}{5}C+32=F$

We can now use the formula $F=\frac{9}{5}C+32$ to find the Fahrenheit temperature when we know the Celsius temperature.

The next example uses the formula for the surface area of a right cylinder.

Example 3

Solve the formula $S=2πr^{2}+2πrh$ for $h$.

Solution

Write the formula.$S=2πr^{2}+2πrh$
Isolate the $h$ term by subtracting $2πr^{2}$ from each side.$S-2πr^{2}=2πr^{2}-2πr^{2} +2πrh$
Simplify.$S-2πr^{2}=2πrh$
Solve for $h$ by dividing both sides by $2πr$.$\frac{S-2πr^{2}}{2πr}=\frac {2πrh}{2πr}$
Simplify.$\frac{S-2πr^{2}}{2πr}=h$

Sometimes we might be given an equation that is solved for $y$ and need to solve it for $x$, or vice versa. In the following example, we’re given an equation with both $x$ and $y$ on the same side and we’ll solve it for $y$.

Example 4

Solve the formula $8x+7y=15$ for $y$.

Solution

We will isolate $y$ on one side of the equation.$8x+7y=15$
Subtract $8x$ from both sides to isolate the term with $y$.$8x-8x+7y=15-8x$
Simplify.$7y=15-8x$
Divide both side by $7$ to make the coefficient of $y$ one.$\frac{7y}{7}=\frac{15-8x}{7}$
Simplify.$y=\frac{15-8x}{7}$

2.3.2 Use Formulas to Solve Geometry Applications

In this objective we will use some common geometry formulas. We will adapt our problem solving strategy so that we can solve geometry applications. The geometry formula will name the variables and give us the equation to solve.

In addition, since these applications will all involve shapes of some sort, most people find it helpful to draw a figure and label it with the given information. We will include this in the first step of the problem solving strategy for geometry applications.

HOW TO: Solve geometry applications.

  1. Read the problem and make sure all the words and ideas are understood.
  2. Identify what you are looking for.
  3. Name what we are looking for by choosing a variable to represent it. Draw the figure and label it with the given information.
  4. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
  5. Solve the equation using good algebra techniques.
  6. Check the answer in the problem and make sure it makes sense.
  7. Answer the question with a complete sentence.

When we solve geometry applications, we often have to use some of the properties of the figures. We will review those properties as needed.

The next examples involves the areas of a triangle. The area of a triangle is one-half base times the height. We can write this as $A=\frac{1}{2}bh$, where $b=$ length of the base and $h=$ height.

The figure is a triangle with its height shown. Its base is b and its height is h. The formula for the area of the triangle is A is equal to one-half times b times h.

Example 5

The area of a triangular painting is $126$ square inches. The base is $18$ inches. What is the height?

Solution

Step 1. Read the problem.
Step 2. Identify what you are looking for.height of a triangle
Step 3. Name.
Choose a variable to represent it. Let $h=$ the height.
Draw the figure and label it with the given information.Area $= 126$ sq. in.
Step 4. Translate.
Write the appropriate formula.$A=\frac{1}{2}bh$
Substitute in the given information.$126=\frac{1}{2} \cdot 18 \cdot h$
Step 5. Solve the equation.$126=9h$
Divide both sides by $9$.$14=h$
Step 6. Check.
$\begin{align*} A&=\frac{1}{2}bh \\ 126&=\frac{1}{2} \cdot 18 \cdot 14 \\ 126 &= 126 \end{align*}$
Step 7. Answer the question.The height of the triangle is 14 inches.

In the next example, we will work with a right triangle. To solve for the measure of each angle, we need to use two triangle properties. In any triangle, the sum of the measures of the angles is $180^{\circ}$. We can write this as a formula: $m∠A+m∠B+m∠C=180$. Also, since the triangle is a right triangle, we remember that a right triangle has one $90^{\circ}$ angle.

Here, we will have to define one angle in terms of another. We will wait to draw the figure until we write expressions for all the angles we are looking for.

Example 6

The measure of one angle of a right triangle is $40$ degrees more than the measure of the smallest angle. Find the measures of all three angles.

Solution

Step 1. Read the problem.
Step 2. Identify what you are looking for.the measures of all three angles
Step 3. Name. Choose a variable to represent it.Let $a=1^{st}$ angle.
$a+40=2^{nd}$ angle
$90=3^{rd}$ angle (the right angle)
Draw the figure and label it with the given information.
Step 4. Translate.
Write the appropriate formula.$m\angle A + m \angle B + m \angle C = 180$
Substitute into the formula.$a+(a+40)+90=180$
Step 5. Solve the equation.$\begin{align*} 2a+130&=180 \\ 2a&=50 \\ a=25 \end{align*}$ first angle

$a+40$ second angle
$25+40$
$65$

$90$ third angle
Step 6. Check.
$\begin{align*} 25+65+90&=180 \\ 180&=180 \end{align*}$
Step 7. Answer the question.The three angles measure $25^{\circ}$, $65^{\circ}$, and $90^{\circ}$.

The next example uses another important geometry formula. The Pythagorean Theorem tells how the lengths of the three sides of a right triangle relate to each other. Writing the formula in every exercise and saying it aloud as you write it may help you memorize the Pythagorean Theorem.

THE PYTHAGOREAN THEOREM

In any right triangle, where $a$ and $b$ are the lengths of the legs, and $c$ is the length of the hypotenuse, the sum of the squares of the lengths of the two legs equals the square of the length of the hypotenuse.

The figure is a right triangle with sides a and b, and a hypotenuse c. a squared plus b squared is equal to c squared. In a right triangle, the sum of the squares of the lengths of the two legs equals the square of the length of the hypotenuse.

We will use the Pythagorean Theorem in the next example.

Example 7

Use the Pythagorean Theorem to find the length of the other leg in

Solution

Step 1. Read the problem.
Step 2. Identify what you are looking for.the length of the leg of the triangle
Step 3. Name.
Choose a variable to represent it.Let $a=$ the leg of the triangle.
Label side $a$.This image has an empty alt attribute; its file name is triangle_a_12_13.jpeg
Step 4. Translate.
Write the appropriate formula.
Substitute.
$\begin{align*} a^{2}+b^{2}&=c^{2} \\ a^{2}+12^{2}&=13^{2} \end{align*}$
Step 5. Solve the equation.
Isolate the variable term.
Use the definition of square root.
Simplify.
$\begin{align*} a^{2}+144&=169 \\ a^{2}&=25 \\ a&=\sqrt{25} \\ a&=5 \end{align*}$
Step 6. Check.
$\begin{align*} 5^{2}+12^{2}&=13^{2} \\ 25+144&=169 \\ 169&=169 \end{align*}$
Step 7. Answer the question.The length of the leg is $5$.

The next example is about the perimeter of a rectangle. Since the perimeter is just the distance around the rectangle, we find the sum of the lengths of its four sides—the sum of two lengths and two widths. We can write is as $P=2L+2W$ where $L$ is the length and $W$ is the width. To solve the example, we will need to define the length in terms of the width.

Example 8

The length of a rectangle is six centimeters more than twice the width. The perimeter is $96$ centimeters. Find the length and width.

Solution

Step 1. Read the problem.
Step 2. Identify what we are looking for.the length and the width
Step 3. Name. Choose a variable to represent the width.
The length is six more than twice the width.
Let $W=$ width.
$2W+6=$ length

This image has an empty alt attribute; its file name is rectangle_w_2wplus6.jpeg

$P=96$ cm
Step 4. Translate.
Write the appropriate formula.$P=2L+2W$
Substitute in the given information.$96=2(2W+6)+2W$
Step 5. Solve the equation.$\begin{align*} 96&=4W=12+2W \\ 96&=6W+12 \\ 84&=6W \\ 14&=W (width)\end{align*}$

$2W+6$ (length)
$2(14)+6$
$34$
Step 6. Check.
$\begin{align*} P&=2L+2W \\ 96&= 2\cdot 34 + 2 \cdot 14 \\ 96&=96 \end{align*}$
Step 7. Answer the question.The length is $34$ cm and the width is $14$ cm.

The next example is about the perimeter of a triangle. Since the perimeter is just the distance around the triangle, we find the sum of the lengths of its three sides. We can write this as $P=a+b+c$, where $a$, $b$, and $c$ are the lengths of the sides.

Example 9

One side of a triangle is three inches more than the first side. The third side is two inches more than twice the first. The perimeter is $29$ inches. Find the length of the three sides of the triangle.

Solution

Step 1. Read the problem.
Step 2. Identify what we are looking for.the lengths of the three sides of a triangle
Step 3. Name. Choose a variable to
represent the length of the first side.
Let $x=$ length of $1^{st}$ side
$x+3 =$ length of $2^{nd}$ side
$2x+2=$ length of $3^{rd}$ side

This image has an empty alt attribute; its file name is triangle_perimeter-29.jpeg
Step 4. Translate.
Write the appropriate formula.
Substitute in the given information.
$P=a+b+c$

$29=x+(x+3)+(2x+2)$
Step 5. Solve the equation.$29=4x+5$
$24=4x$
$6=x$ length of first side

$x+3$ length of second side
$6+3$
$9$

$2x+2$ length of third side
$2 \cdot 6 +2$
$14$
Step 6. Check.
This image has an empty alt attribute; its file name is triangle_6_9_14.jpeg

$\begin{align*} 29&=6+9+14 \\ 29&+29 \end{align*}$
Step 7. Answer the question.The lengths of the sides of the triangle
are $6$, $9$, and $14$ inches.

Example 10

The perimeter of a rectangular soccer field is $360$ feet. The length is $40$ feet more than the width. Find the length and width.

Solution

Step 1. Read the problem.
Step 2. Identify what we are looking for.the length and width of the soccer field
Step 3. Name. Choose a variable to represent it.
The length is $40$ feet more than the width.
Draw the figure and label it with the
given information.
Let $w=$ width.
$w+40=$ length

This image has an empty alt attribute; its file name is soccer_field_360_perimeter.jpeg
Step 4. Translate.
Write the appropriate formula and
substitute.
$P=2L+2W$

$360=2(w+40)+2w$
Step 5. Solve the equation.$360=2w+80+2w$
$360=4w+80$
$280=4w$
$70=w$ the width of the field

$w+40$ length of the field
$70+40$
$110$
Step 6. Check.
$\begin{align*} P&=2L+2W \\ 360&=2(110)+2(70) \\ 360&=360 \end{align*}$
Step 7. Answer the question.The length of the soccer field is $110$ feet
and the width is $70$ feet.

Applications of these geometric properties can be found in many everyday situations as shown in the next example.

Example 11

Kelvin is building a gazebo and wants to brace each corner by placing a $10$ inch piece of wood diagonally as shown.

How far from the corner should he fasten the wood if wants the distances from the corner to be equal? Approximate to the nearest tenth of an inch.

Solution

Step 1. Read the problem.
Step 2. Identify what we are looking for.the distance from the corner that the
bracket should be attached
Step 3. Name. Choose a variable to represent it.
Draw the figure and label it with the given
information.
Let $x=$ the distance from the corner.

This image has an empty alt attribute; its file name is right_triangle_hypotenuse_10.jpeg
Step 4. Translate.
Write the appropriate formula and substitute.
$a^{2}+b^{2}=c^{2}$
$x^{2}+x^{2}=10^{2}$
Step 5. Solve the equation.
Isolate the variable.
Use the definition of square root.
Simplify. Approximate to the nearest tenth.
$\begin{align*} 2x^2&=100 \\ x^{2}&=50 \\ x&=\sqrt{50} \\ x&\approx 7.1 \end{align*}$
Step 6. Check.
$\begin{align*} a^{2}+b^{2}&=c^{2} \\ (7.1)^{2}+(7.1)^{2}&\approx 10^{2} \end{align*}$ Yes.
Step 7. Answer the question.Kelvin should fasten each piece of wood
approximately $7.1$ inches from the corner.
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