Algebra Foundations for Normal People by Gabriel Beguerie - HTML preview
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Chapter goals
By the end of this chapter, you will be able to:
• read points in the coordinate plane,
• compute slope from two points,
• interpret slope as rate of change,
• identify the slope and intercept in the equation of a line,
• build a line equation when you know a slope and one point.
Prerequisites (if this is hard, go back to Unit 0)
You need signed numbers, fractions, and basic equations. Slope is a fraction: change in y over change in x.
Why this matters
Key idea
This chapter is the doorway from algebra into functions. A line is one of the first full mathematical models students learn to read, calculate with, and graph.
Real life: what it’s for
Slope appears in ramps, roads, stairs, price-per-unit models, speed graphs, and trend lines.
Whenever one quantity changes in response to another, slope is nearby.
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Ordered pairs and the coordinate plane
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Ordered pairs and the coordinate plane
Key idea
An ordered pair ( 𝑥, 𝑦) tells you a horizontal position and a vertical position. The first number is the 𝑥-coordinate. The second number is the 𝑦-coordinate. Order matters. So (3 , −2) and (−2 , 3) are different points.
Worked example (no skipped steps)
What does the point (3 , −2) mean?
Step 1: start at the origin. The origin is (0 , 0).
Step 2: move 3 units to the right because the 𝑥-coordinate is 3.
Step 3: move 2 units down because the 𝑦-coordinate is −2.
That location is (3 , −2).
𝑦
2
1
𝑥
-1
1
righ 2
t 3 3
4
-1
down 2
-2
(3 , −2)
-3
Slope as rate of change
Key idea
Slope means
change in 𝑦
slope = change in 𝑥
People also call this rise over run. The rise is the vertical change. The run is the horizontal change.
If slope is positive, the line rises as you move right. If slope is negative, the line falls as you move right. If slope is 0, the line is flat.
Recipe (steps that always work)
Given two points ( 𝑥 1 , 𝑦 1) and ( 𝑥 2 , 𝑦 2), the slope is 𝑦 2 − 𝑦 1
𝑚 = 𝑥 2 − 𝑥 1
provided 𝑥 2 − 𝑥 1 ̸= 0.
Pick one subtraction order and keep it consistent in both the numerator and denominator. If you switch only one of them, you change the answer.
Chapter 11. Coordinates, slope, and straight lines
Worked example (no skipped steps)
Find the slope through (1 , 2) and (5 , 10).
Step 1: label the coordinates.
𝑥 1 = 1 ,
𝑦 1 = 2 ,
𝑥 2 = 5 ,
𝑦 2 = 10
Step 2: substitute into the slope formula.
𝑦 2 − 𝑦 1
10 − 2
𝑚 =
=
𝑥 2 − 𝑥 1
5 − 1
Step 3: simplify the numerator and denominator.
10 − 2 = 8 ,
5 − 1 = 4
So:
8
𝑚 =
= 2
4
Answer: the slope is 2.
Worked example (no skipped steps)
Find the slope through (2 , 5) and (6 , 1).
Step 1: substitute into the formula.
1 − 5
𝑚 = 6 − 2
Step 2: simplify top and bottom.
1 − 5 = −4 ,
6 − 2 = 4
So:
−4
𝑚 =
= −1
4
Answer: the slope is −1.
Worked example (no skipped steps)
Find the slope through (−1 , 4) and (3 , 4).
Step 1: use the formula.
4 − 4
𝑚 = 3 − (−1)
Step 2: simplify carefully.
4 − 4 = 0 ,
3 − (−1) = 3 + 1 = 4
So:
0
𝑚 =
= 0
4
Answer: the line is horizontal, so its slope is 0.
Undefined slope and vertical lines
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Undefined slope and vertical lines
Key idea
A vertical line has undefined slope because its change in 𝑥 is 0. That would make the slope formula divide by 0, and division by 0 is not allowed.
Worked example (no skipped steps)
What happens to the slope through (7 , −2) and (7 , 5)?
Use the formula:
5 − (−2)
𝑚 =
7 − 7
Simplify the numerator:
5 − (−2) = 7
Simplify the denominator:
7 − 7 = 0
So the slope would be:
7
0
That is undefined because division by 0 is not allowed.
Answer: the line is vertical, so its slope is undefined.
Equation of a line
Key idea
A common form is
𝑦 = 𝑚𝑥 + 𝑏
where 𝑚 is the slope and 𝑏 is the 𝑦-intercept.
The 𝑦-intercept is the value of 𝑦 when 𝑥 = 0. So it is the point where the line crosses the 𝑦-axis.
Worked example (no skipped steps)
In the line
𝑦 = 3 𝑥 − 4 ,
identify the slope and the 𝑦-intercept.
Compare the equation to
𝑦 = 𝑚𝑥 + 𝑏.
Here,
𝑚 = 3 ,
𝑏 = −4 .
Answer: the slope is 3 and the 𝑦-intercept is −4.
Chapter 11. Coordinates, slope, and straight lines
Worked example (no skipped steps)
In the line
𝑦 = −2 𝑥 + 7 ,
identify the slope and the 𝑦-intercept.
Compare to
𝑦 = 𝑚𝑥 + 𝑏.
So:
𝑚 = −2 ,
𝑏 = 7 .
Answer: the slope is −2 and the 𝑦-intercept is 7.
Note
A common beginner mistake is to think “the line goes through ( 𝑚, 𝑏).” That is not what 𝑦 = 𝑚𝑥 + 𝑏 says. It says:
• 𝑚 controls the steepness,
• 𝑏 tells where the line crosses the 𝑦-axis.
Build a line when you know the slope and one point
Key idea
If you know the slope and one point, you can still use
𝑦 = 𝑚𝑥 + 𝑏.
The slope is already known. The point lets you solve for 𝑏.
Worked example (no skipped steps)
Write the equation of the line with slope 2 and 𝑦-intercept −3.
Use the form
𝑦 = 𝑚𝑥 + 𝑏
Substitute 𝑚 = 2 and 𝑏 = −3:
𝑦 = 2 𝑥 − 3
Answer: 𝑦 = 2 𝑥 − 3.
Build a line when you know the slope and one point 95
Worked example (no skipped steps)
Write the equation of the line through (1 , 4) with slope −2 in the form 𝑦 = 𝑚𝑥 + 𝑏.
Step 1: start with the slope-intercept form.
𝑦 = 𝑚𝑥 + 𝑏
Step 2: substitute the known slope 𝑚 = −2.
𝑦 = −2 𝑥 + 𝑏
Step 3: use the point (1 , 4). That means when 𝑥 = 1, the value of 𝑦 is 4. So substitute 𝑥 = 1
and 𝑦 = 4:
4 = −2(1) + 𝑏
Step 4: simplify.
4 = −2 + 𝑏
Step 5: solve for 𝑏. Add 2 to both sides: 6 = 𝑏
So:
𝑏 = 6
Step 6: put that value back into the line equation.
𝑦 = −2 𝑥 + 6
Answer: the equation is
𝑦 = −2 𝑥 + 6 .
Worked example (no skipped steps)
Does the point (2 , 5) lie on the line 𝑦 = 3 𝑥 − 1?
Step 1: substitute 𝑥 = 2 into the line.
𝑦 = 3(2) − 1 = 6 − 1 = 5
Step 2: compare the result with the given 𝑦-value. The point says 𝑦 = 5, and the line also gives 𝑦 = 5.
Answer: yes, the point (2 , 5) lies on the line.
Bridge (why this matters later)
The next book begins to ask deeper questions about graphs, transformations, and function behavior. This chapter makes that jump much smoother because lines are the first complete function family most students learn to read with confidence.
Chapter 11. Coordinates, slope, and straight lines
Common mistakes (and how to avoid them)
• switching the order of subtraction in the numerator but not the denominator,
• confusing a point with a slope,
• forgetting that horizontal lines have slope 0,
• dividing by zero when the line is vertical,
• trying to write a line from a point and slope without first finding 𝑏.
Recap: what you’re taking away
A line connects three views at once:
• a picture on the coordinate plane,
• a rate of change through slope,
• an equation such as 𝑦 = 𝑚𝑥 + 𝑏.
If those three views feel connected, you are in good shape for Math Functions for Normal People.
Problems
11.1. Explain what the point (3 , −2) means.
11.2. Find the slope through (1 , 2) and (5 , 10).
11.3. Find the slope through (2 , 5) and (6 , 1).
11.4. Find the slope through (−1 , 4) and (3 , 4).
11.5. In 𝑦 = 3 𝑥 − 4, identify 𝑚 and 𝑏.
11.6. In 𝑦 = −2 𝑥 + 7, identify 𝑚 and 𝑏.
11.7. Decide whether a line with slope 0 rises, falls, or stays flat.
11.8. Explain why a vertical line has no slope.
11.9. Find the slope through (−2 , 4) and (3 , −6).
11.10. Write the equation of the line with slope 2 and 𝑦-intercept −3.
11.11. Write the equation of the line through (1 , 4) with slope −2.
11.12. Decide whether the point (2 , 5) lies on the line 𝑦 = 3 𝑥 − 1.
11.13. Find the slope through (2 , 5) and (6 , 13).
11.14. Find the slope through (−4 , −1) and (2 , 2).
11.15. Find the slope through (7 , −2) and (7 , 5).
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11.16. Write the equation of the line with slope 3 and 𝑦-intercept −2 in the form 𝑦 = 𝑚𝑥 + 𝑏.
11.17. Write the equation of the line with slope −1 passing through (1 , 2) in the form 𝑦 = 𝑚𝑥 + 𝑏.
11.18. For the line 𝑦 = 2 𝑥 − 5, find 𝑦 when 𝑥 = 4.
11.19. For the line 𝑦 = 2 𝑥 − 5, find 𝑥 when 𝑦 = 9.
11.20. Does the point (3 , 1) lie on the line 𝑦 = 2 𝑥 − 5?
Quick checklist (if you miss it, reread and try again)
• I can read an ordered pair.
• I know slope as change in 𝑦 over change in 𝑥.
• I can identify slope and intercept in 𝑦 = 𝑚𝑥 + 𝑏.
• I can build a line equation when I know a slope and one point.
Formulas and rearranging variables