Section3.3Isolating a Linear Variable
¶In this section, we will learn how to solve linear equations and inequalities with more than one variables.
Subsection3.3.1Solving for a Variable
The formula of calculating a rectangle's area is \(A=lw\text{,}\) where \(l\) stands for the rectangle's length, and \(w\) stands for width. When a rectangle's length and width are given, we can easily calculate its area.
What if a rectangle's area and length are given, and we need to calculate its width?
If a rectangle's area is given as 12 m^{2}, and it's length is given as 4 m, we could find its width this way:
If we need to do this many times, we would love to have an easier way, without solving an equation each time. We will solve for \(w\) in the formula \(A=lw\text{:}\)
Now if we want to find the width when \(l=4\) is given, we can simply replace \(l\) with \(4\) and simplify.
We solved for \(w\) in the formula \(A=lw\) once, and we could use the new formula \(w=\frac{A}{l}\) again and again saving us a lot of time down the road. Let's look at a few examples.
Remark3.3.2
Note that in solving for \(A\text{,}\) we divided each side of the equation by \(l\text{.}\) The operations that we apply, and the order in which we do them, are determined by the operations in the original equation. In the original equation \(A=lw\text{,}\) we saw that \(w\) was multiplied by \(l\text{,}\) and so we knew that in order to “undo” that operation, we would need to divide each side by \(l\text{.}\) We will see this process of “undoing” the operations throughout this section.
Example3.3.3
Solve for \(R\) in \(P=RC\text{.}\) (This is the relationship between profit, revenue, and cost.)
To solve for \(R\text{,}\) we first want to note that \(C\) is subtracted from \(R\text{.}\) To “undo” this, we will need to add \(C\) to each side of the equation:
Example3.3.4
Solve for \(x\) in \(y=mx+b\text{.}\) (This is a line's equation in slopeintercept form.)
In the equation \(y=mx+b\text{,}\) we see that \(x\) is multiplied by \(m\) and then \(b\) is added to that. Our first step will be to isolate \(mx\text{,}\) which we'll do by subtracting \(b\) from each side of the equation:
Now that we have \(mx\) on it's own, we'll note that \(x\) is multiplied by \(m\text{.}\) To “undo” this, we'll need to divide each side of the equation by \(m\text{:}\)
Warning3.3.5
It's important to note in Example 3.3.4 that each side was divided by \(m\text{.}\) We can't simply divide \(y\) by \(m\text{,}\) as the equation would no longer be equivalent.
Example3.3.6
Solve for \(b\) in \(A=\frac{1}{2}bh\text{.}\) (This is the area formula for a triangle.)
To solve for \(b\text{,}\) we need to determine what operations need to be “undone.” The expression \(\frac{1}{2}bh\) has multiplication between \(\frac{1}{2}\) and \(b\) and \(h\text{.}\) As a first step, we will multiply each side of the equation by \(2\) in order to eliminate the denominator of \(2\text{:}\)
As a last step, we will “undo” the multiplication between \(b\) and \(h\) by dividing each side by \(h\text{:}\)
Example3.3.7
Solve for \(y\) in \(2x+5y=10\text{.}\) (This is a linear equation in standard form.)
To solve for \(y\) in the equation \(2x+5y=10\text{,}\) we will first have to solve for \(5y\text{.}\) We'll do so by subtracting \(2x\) from each side of the equation. After that, we'll be able to divide each side by \(5\) to finish solving for \(y\text{:}\)
Remark3.3.8
As we will learn in later sections, the result in Example 3.3.7 can also be written as \(y=\frac{10}{5}\frac{2x}{5}\) which can then be written as \(y=2\frac{2}{5}x\text{.}\)
Example3.3.9
Solve for \(F\) in \(C=\frac{5}{9}(F32)\text{.}\) (This represents the relationship between temperature in degrees Celsius and degrees Fahrenheit.)
To solve for \(F\text{,}\) we first need to see that it is contained inside a set of parentheses. To get the expression \(F32\) on its own, we'll need to eliminate the \(\frac{5}{9}\) outside those parentheses. One way we can “undo” this multiplication is by dividing each side by \(\frac{5}{9}\text{.}\) As we learned in Section 3.2 though, a better approach is to instead multiply each side by the reciprocal of \(\frac{9}{5}\text{:}\)
Now that we have \(F32\text{,}\) we simply need to add \(32\) to each side to finish solving for \(F\text{:}\)
SubsectionExercises
Solving for a Variable
1

Solve this linear equation for \(t\text{.}\)
\(t+7=13\)

Solve this linear equation for \(y\text{.}\)
\(y + x = B\)
2

Solve this linear equation for \(t\text{.}\)
\(t+7=9\)

Solve this linear equation for \(x\text{.}\)
\(x + c = b\)
3

Solve this linear equation for \(x\text{.}\)
\(x1=7\)

Solve this linear equation for \(r\text{.}\)
\(rp=7\)
4

Solve this linear equation for \(x\text{.}\)
\(x9=1\)

Solve this linear equation for \(y\text{.}\)
\(yt=1\)
5

Solve this linear equation for \(y\text{.}\)
\(y+7=1\)

Solve this linear equation for \(x\text{.}\)
\(x+C=A\)
6

Solve this linear equation for \(y\text{.}\)
\(y+1=3\)

Solve this linear equation for \(r\text{.}\)
\(r+t=C\)
7

Solve this linear equation for \(r\text{.}\)
\(7r = 42\)

Solve this linear equation for \(y\text{.}\)
\(Ay=q\)
8

Solve this linear equation for \(r\text{.}\)
\(7r = 14\)

Solve this linear equation for \(t\text{.}\)
\(qt=C\)
9

Solve this linear equation for \(t\text{.}\)
\(\frac{t}{7}=8\)

Solve this linear equation for \(r\text{.}\)
\(\frac{r}{b}=B\)
10

Solve this linear equation for \(t\text{.}\)
\(\frac{t}{3}=8\)

Solve this linear equation for \(y\text{.}\)
\(\frac{y}{m}=B\)
11

Solve this linear equation for \(t\text{.}\)
\(9t+4=49\)

Solve this linear equation for \(x\text{.}\)
\(px+B=A\)
12

Solve this linear equation for \(x\text{.}\)
\(7x+5=40\)

Solve this linear equation for \(r\text{.}\)
\(Br+p=n\)
13

Solve this linear equation for \(x\text{.}\)
\(xy = a\)

Solve this linear equation for \(y\text{.}\)
\(xy = a\)
14

Solve this linear equation for \(y\text{.}\)
\(yt = c\)

Solve this linear equation for \(t\text{.}\)
\(yt = c\)
15

Solve this linear equation for \(y\text{.}\)
\(y+r = p\)

Solve this linear equation for \(r\text{.}\)
\(y+r = p\)
16

Solve this linear equation for \(r\text{.}\)
\(r+x = a\)

Solve this linear equation for \(x\text{.}\)
\(r+x = a\)
17

Solve this linear equation for \(C\text{.}\)
\(pr+C=A\)

Solve this linear equation for \(p\text{.}\)
\(pr+C=A\)
18

Solve this linear equation for \(m\text{.}\)
\(Ax+m=p\)

Solve this linear equation for \(A\text{.}\)
\(Ax+m=p\)
19

Solve this linear equation for \(p\text{.}\)
\(y=rp+A\)

Solve this linear equation for \(r\text{.}\)
\(y=rp+A\)
20

Solve this linear equation for \(q\text{.}\)
\(t=nq+b\)

Solve this linear equation for \(n\text{.}\)
\(t=nq+b\)
21
Solve this linear equation for \(x\text{:}\)
\(\displaystyle{ y=mxb}\)
22
Solve this linear equation for \(x\text{:}\)
\(\displaystyle{ y=mx+b}\)
23

Solve this equation for \(b\text{:}\)
\(18=\frac{1}{2} b \cdot 4\)

Solve this equation for \(b\text{:}\)
\(A=\frac{1}{2} b \cdot h\)
24

Solve this equation for \(b\text{:}\)
\(18=\frac{1}{2} b \cdot 6\)

Solve this equation for \(b\text{:}\)
\(A=\frac{1}{2} b \cdot h\)
25
Solve this linear equation for \(r\text{:}\)
\(\displaystyle{ C=2 \pi r }\)
26
Solve this linear equation for \(h\text{:}\)
\(\displaystyle{ V= \pi r^{2} h }\)
27
Solve these linear equations for \(t\text{.}\)
\(\frac{t}{3}+5=8\)
\(\frac{t}{y}+5=p\)
28
Solve these linear equations for \(t\text{.}\)
\(\frac{t}{3}+2=3\)
\(\frac{t}{x}+2=a\)
29
Solve this linear equation for \(t\text{:}\)
\(\displaystyle{ \frac{t}{r}+C=p }\)
30
Solve this linear equation for \(x\text{:}\)
\(\displaystyle{ \frac{x}{y}+B=a }\)
31
Solve this linear equation for \(x\text{:}\)
\(\displaystyle{ \frac{x}{5}+r=a }\)
32
Solve this linear equation for \(y\text{:}\)
\(\displaystyle{ \frac{y}{2}+r=q }\)
33
Solve this linear equation for \(b\text{:}\)
\(\displaystyle{ A=m\frac{8b}{a} }\)
34
Solve this linear equation for \(A\text{:}\)
\(\displaystyle{ c=n\frac{4A}{q} }\)
35
Solve this linear equation for \(x\text{:}\)
\(\displaystyle{ Ax+By=C }\)
Note that the variables are upper case A, B, and C and lower case x and y.
36
Solve this linear equation for \(y\text{:}\)
\(\displaystyle{ Ax+By=C }\)
Note that the variables are upper case A, B, and C and lower case x and y.