Compare linear equations, graphs, and tables

key notes:

What is a Linear Relationship?

A linear relationship shows a constant rate of change between two variables.
It can be represented in three main ways:

1. Equation – Algebraic form (e.g., y=2x+3)
2. Graph – A straight line on a coordinate plane
3. Table – A set of ordered pairs (x,y) showing the relationship

---

Forms of a Linear Equation

• Slope-Intercept Form:

y=mx+b

• mmm = slope (rate of change)
• bbb = y-intercept (where the line crosses the y-axis)

Standard Form:

Ax+By=C

Can be converted to slope-intercept form to compare easily

---

How to Compare Using Equations, Graphs, and Tables

A. Compare Slope (m)

• In equation: Look directly at mmm in

y=mx+b

• In graph: Choose two points, use formula:

m=change in y/change in x – y₂-y₁/ x_2-x₁

• In table: Check how much y changes when x increases by 1.

---

B. Compare Y-Intercept (b)

• In equation: Look directly at bbb.
• In graph: Find where the line crosses y-axis (x = 0).
• In table: Look for the value of y when x = 0.

---

Worked Examples

Example 1 – Compare from Equations

Equation 1: y=2x+5
Equation 2: y=2x−3

• Slopes: both m=2→ lines are parallel (same steepness).
• Intercepts: 5 vs -3 → different starting points on y-axis.

✅ Lines have same slope, different intercepts → parallel lines.

---

Example 2 – Compare from Graphs

If one line rises faster than another when moving left to right, it has a larger slope.
If lines cross y-axis at different points, they have different y-intercepts.

---

Example 3 – Compare from Tables

x	y₁	y₂
0	4	1
1	6	3
2	8	5

• Slope for y₁​: increase by 2 each time x increases by 1 → slope = 2.
• Slope for y₂​: also increase by 2 → slope = 2.
• Intercept for y₁ = 4, for y₂= 1.

✅ Same slope, different intercepts → parallel.

---

Common Observations When Comparing

• Same slope, different intercept → parallel lines.
• Different slope → lines intersect at one point.
• Same slope and intercept → same line (overlapping).

---

Practice Questions

• Compare y=3x+1 and y=3x−4
• Compare the lines with points (0, 2) & (2, 6) and (0, 1) & (2, 7).
• From the table:

| x | y₁ | y₂ |
| – | — | — |
| 0 | 5 | 7 |
| 1 | 8 | 9 |
| 2 | 11 | 11 |
Compare slopes and intercepts.

Learn with an example

Let’s Practice!