8 Applications of Linear Equations: Workbook Examples/Practice
3.5 Constructing Linear Equations
Quite often, students are required to find the equation of a line given only a data point and a slope or two data points. The simpler of these problems is to find the equation when given a slope and a data point. To do this, use the equation that defines the slope of a line:
This becomes:
Examples/Practice
Write the slope-intercept form of each linear equation using the given point and slope.
1.) 
and 
2.) 
and 
3.) 
and 
4.) 
and 
3.6 Perpendicular and Parallel Lines
Perpendicular, parallel, horizontal, and vertical lines are special lines that have properties unique to each type. Parallel lines, for instance, have the same slope, whereas perpendicular lines are the opposite and have negative reciprocal slopes. Vertical lines have a constant 
-value, and horizontal lines have a constant 
-value.
Two equations govern perpendicular and parallel lines:
For parallel lines, the slope of the first line is the same as the slope for the second line. If the slopes of these two lines are called 
and 
, then 
.
Perpendicular lines are slightly more difficult to understand. If one line is rising, then the other must be falling, so both lines have slopes going in opposite directions. Thus, the slopes will always be negative to one another. The other feature is that the slope at which one is rising or falling will be exactly flipped for the other one. This means that the slopes will always be negative reciprocals to each other. If the slopes of these two lines are called and , then 
.
Examples/Practice
Find the slope of any line that would be parallel to each given line.
1.) 
2.) 
3.) 
4.) 
Find the slope of any line that would be perpendicular to each given line.
5.) 
6.) 
3.7 Numeric Word Problems
Number-based word problems can be very confusing, and it takes practice to convert a word-based sentence into a mathematical equation. The best strategy to solve these problems is to identify keywords that can be pulled out of a sentence and use them to set up an algebraic equation.
Variables that are to be solved for are often written as “a number,” “an unknown,” or “a value.”
Examples/Practice
Write the formula defining each relationship. Do not solve.
1.) Five more than twice an unknown number is 25.
2.) When an unknown number is decreased by 8, the difference is half the unknown number.
3.) The sum of three consecutive integers is 21.
4.) The sum of the first two of three odd consecutive integers, less the third, is 5.
Write and solve the equation describing each relationship.
5.) When five is added to three times a certain number, the result is 17. What is the number?
6.) Eleven less than seven times a number is five more than six times the number. Find the number.
7.) The sum of three consecutive integers is −126. What are the integers?
8.) Find three consecutive odd integers such that the sum of the first, two times the second, and three times the third is 70.