Writing from Slope and Y-Intercept

Write the equation of a line with slope $m = 3$ and y-intercept $b = -2$.

Identify the slope: $m = 3$ = Slope is 3

Identify the y-intercept: $b = -2$ = Y-intercept is $-2$

Substitute into slope-intercept form: $y = mx + b$ $y = 3x + (-2)$ = $y = 3x - 2$

Answer: $y = 3x - 2$

Writing from a Point and Slope

Write the equation of a line with slope $m = 2$ that passes through the point $(3, 7)$.

Write point-slope form: $y - y_1 = m(x - x_1)$ = Template ready

Substitute the values: $y - 7 = 2(x - 3)$ = Point-slope form

Distribute the slope: $y - 7 = 2x - 6$ = Simplified right side

Solve for y: $y = 2x - 6 + 7$ $y = 2x + 1$ = $y = 2x + 1$

Answer: $y = 2x + 1$

Writing from Two Points

Write the equation of the line passing through $(1, 2)$ and $(4, 11)$.

Calculate the slope: $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{11 - 2}{4 - 1} = \frac{9}{3}$ = $m = 3$

Use point-slope form with $(1, 2)$: $y - 2 = 3(x - 1)$ = Point-slope equation

Distribute: $y - 2 = 3x - 3$ = Expanded form

Solve for y: $y = 3x - 3 + 2$ $y = 3x - 1$ = $y = 3x - 1$

Answer: $y = 3x - 1$

Writing from a Table

Write the equation for this table: | $x$ | 0 | 1 | 2 | 3 | |-----|---|---|---|---| | $y$ | 4 | 7 | 10 | 13 |

Find the rate of change (slope): When $x$ increases by 1, $y$ increases by 3 $m = \frac{7-4}{1-0} = 3$ = $m = 3$

Find the y-intercept: When $x = 0$, $y = 4$ $b = 4$ = $b = 4$

Write the equation: $y = mx + b$ $y = 3x + 4$ = $y = 3x + 4$

Verify with another point: Check $(2, 10)$: $3(2) + 4 = 10$ ✓ = Equation confirmed!

Answer: $y = 3x + 4$

Writing from a Word Problem

A taxi charges a 3 dollar base fee plus 2 dollars per mile. Write an equation for the total cost $C$ based on miles $m$.

Identify the rate of change: Cost increases by 2 dollars per mile = Slope $= 2$

Identify the starting value: Base fee is 3 dollars (when $m = 0$) = Y-intercept $= 3$

Write the equation: $C = 2m + 3$ = Total cost equation

Interpret: For 5 miles: $C = 2(5) + 3 = 13$ dollars = 13 dollars for 5 miles

Answer: $C = 2m + 3$

Mistake: Confusing slope and y-intercept positions

Why: In $y = mx + b$, the coefficient of $x$ is ALWAYS the slope, and the constant is ALWAYS the y-intercept.

Correct: Remember: $m$ multiplies $x$ (slope), $b$ stands alone (y-intercept). In $y = 5 + 3x$, the slope is 3, not 5!

Mistake: Calculating slope as $\frac{x_2 - x_1}{y_2 - y_1}$

Why: This gives the reciprocal of the correct slope!

Correct: Slope is always $\frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}$ (y's on top, x's on bottom)

Mistake: Forgetting to distribute negative slopes

Why: When $m = -2$ and the point is $(3, 5)$: $y - 5 = -2(x - 3)$ becomes $y - 5 = -2x + 6$, not $-2x - 6$

Correct: A negative times a negative is positive: $-2 \times (-3) = +6$

Mistake: Using inconsistent points in slope formula

Why: Switching the order of points mid-calculation gives wrong answers.

Correct: If you start with $(x_1, y_1) = (2, 5)$, use that consistently: $m = \frac{y_2 - 5}{x_2 - 2}$

Cell Phone Plans

Phone companies use linear equations to calculate monthly bills with base fees and per-unit charges.

A plan costs 30 dollars per month plus 0.10 dollars per text. The equation is $C = 0.10t + 30$.

Science Experiments

Scientists use linear equations to model relationships like temperature changes or distance traveled.

Water cools by 5 degrees per minute from an initial 80 degrees: $T = -5t + 80$

Savings Goals

Linear equations help track savings progress toward financial goals.

Starting with 200 dollars and saving 50 dollars weekly: $S = 50w + 200$

Q: Which form should I use - slope-intercept or point-slope?

A: If you know the y-intercept, use slope-intercept ($y = mx + b$). If you only know a point that's not on the y-axis, point-slope ($y - y_1 = m(x - x_1)$) is easier. Both give the same line!

Q: What if I get a different equation using a different point?

A: Different points give equations that look different but represent the same line. For example, using points $(1, 5)$ and $(2, 8)$ with slope 3: $y - 5 = 3(x - 1)$ and $y - 8 = 3(x - 2)$ both simplify to $y = 3x + 2$.

Q: How do I know if my equation is correct?

A: Substitute a known point into your equation. If both sides are equal, you're correct! For example, if your line passes through $(2, 7)$ and your equation is $y = 2x + 3$, check: $7 = 2(2) + 3 = 7$ ✓