1.
Consider the equation \(A = \frac{BC + D}{E}\text{.}\)
- Solve for \(D\text{.}\)
- Solve for \(B\text{.}\)
- Solve for \(E\text{.}\)
- \(\displaystyle D=AE-BC\)
- \(\displaystyle B=\frac{AE-D}{C}\)
- \(\displaystyle E=\frac{BC+D}{A}\)
Section 1.2 Algebra
Subsection 1.2.1 Basics for lines
Let \(L\) be a line in the \(x,y\)-plane that is not vertical (that is, not parallel to the \(y\)-axis). Visualize traveling along the line \(L\) from a starting point \((A,B)\) to a final point \((P,Q)\text{.}\) From start to finish, your position along the \(x\)-axis will change from \(A\) to \(P\text{.}\) The deviation of \(P\) from \(A\) (also called the horizontal displacement from \((A,B)\) to \((P,Q)\)) is called the run of your trip.
\begin{equation*}
\text{ run } = P-A
\end{equation*}
Likewise, your position along the \(y\)-axis will change from \(B\) to \(Q\text{.}\) The deviation of \(Q\) from \(B\) (also called the vertical displacement) is called the rise of your trip. See Figure Figure 1.2.1.
\begin{equation*}
\text{ rise } = Q-B
\end{equation*}
The fraction rise/run is called the slope of the line \(L\text{.}\)
\begin{equation}
\text{ slope } = \frac{ \text{ rise }}{\text{ run }}=\frac{Q-B}{P-A}\tag{1.2.1}
\end{equation}
Solving (1.2.1) for rise, we have
\begin{equation*}
\text{ rise } = { \text{ (slope)}}{\text{(run) }}
\end{equation*}
and solving (1.2.1) for \(Q\text{,}\) we have
\begin{equation}
Q=\text{ (slope)(run)} + B = \text{ (slope)}(P-A)+B.\tag{1.2.2}
\end{equation}
Here is a basic problem involving lines: You are given a line \(L\text{,}\) the slope \(m\) of \(L\text{,}\) a starting point \((A,B)\) on \(L\text{,}\) and the \(x\)-coordinate \(P\) of a final point \((P,Q)\) on \(L\text{.}\) Your task is to find \(Q\text{.}\)
The solution uses (1.2.2) broken into three these steps:
- Find the run using \(\text{ run } = P-A\text{.}\)
- Find the rise using \(\text{ rise } = \text{ (slope)(run)}\text{.}\)
- Find \(Q\) using \(Q=\text{ rise } + B\text{.}\)
Exercises 1.2.2 Algebra practice problems
2.
Names of quantities in the top row of the table below match Subsection 1.2.1. Find the values of run, rise, and \(Q\) for each row of the table.
\begin{align*}
A
\amp\spacer\amp B
\amp\spacer\amp
\text{slope}
\amp\spacer\amp P\\
\rule{.5in}{.1ex} \amp \amp \rule{.5in}{.1ex} \amp \amp
\rule{.5in}{.1ex} \amp \amp \rule{.5in}{.1ex}\\
20.13
\amp\spacer\amp 15.86
\amp\spacer\amp 1.1
\amp\spacer\amp 24.8\\
20.13
\amp\spacer\amp 15.86
\amp\spacer\amp 1.1
\amp\spacer\amp 17.5\\
103.20
\amp\spacer\amp 26.78
\amp\spacer\amp -0.65
\amp\spacer\amp 110.0\\
103.20
\amp\spacer\amp 26.78
\amp\spacer\amp -0.65
\amp\spacer\amp 95.0
\end{align*}
\begin{align*}
A
\amp\spacer\amp B
\amp\spacer\amp
\text{slope}
\amp\spacer\amp P
\amp\spacer\amp \text{run}
\amp\spacer\amp \text{rise} \amp\spacer\amp Q\\
\rule{.5in}{.1ex}
\amp \amp \rule{.5in}{.1ex}
\amp \amp \rule{.5in}{.1ex}
\amp \amp \rule{.5in}{.1ex}
\amp \amp \rule{.5in}{.1ex}
\amp \amp \rule{.5in}{.1ex}
\amp \amp \rule{.5in}{.1ex} \\
20.13
\amp\spacer\amp 15.86
\amp\spacer\amp 1.1
\amp\spacer\amp 24.8
\amp\spacer\amp 4.67
\amp\spacer\amp 5.14
\amp\spacer\amp 20.10\\
20.13
\amp\spacer\amp 15.86
\amp\spacer\amp 1.1
\amp\spacer\amp 17.5
\amp\spacer\amp -2.63
\amp\spacer\amp -2.89
\amp\spacer\amp 12.97\\
103.20
\amp\spacer\amp 26.78
\amp\spacer\amp -0.65
\amp\spacer\amp 110.0
\amp\spacer\amp 6.80
\amp\spacer\amp -4.42
\amp\spacer\amp 22.36\\
103.20
\amp\spacer\amp 26.78
\amp\spacer\amp -0.65
\amp\spacer\amp 95.0
\amp\spacer\amp -8.20
\amp\spacer\amp 5.33
\amp\spacer\amp 32.11
\end{align*}
