Writing an Equation from Slope and Y-Intercept

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

Identify the form to use: We have slope and y-intercept, so use slope-intercept form: $y = mx + b$ = Use $y = mx + b$

Substitute the slope: $m = 3$, so $y = 3x + b$ = $y = 3x + b$

Substitute the y-intercept: $b = -2$, so $y = 3x + (-2)$ = $y = 3x - 2$

Answer: $y = 3x - 2$

Writing an Equation from a Point and Slope

Write the equation of a line passing through $(2, 5)$ with slope $-4$.

Choose the appropriate form: We have a point and slope, so use point-slope form: $y - y_1 = m(x - x_1)$ = Use point-slope form

Identify the values: $(x_1, y_1) = (2, 5)$ and $m = -4$ = $x_1 = 2$, $y_1 = 5$, $m = -4$

Substitute into the formula: $y - 5 = -4(x - 2)$ = $y - 5 = -4(x - 2)$

Convert to slope-intercept form (optional): $y - 5 = -4x + 8$, so $y = -4x + 13$ = $y = -4x + 13$

Answer: $y - 5 = -4(x - 2)$ or $y = -4x + 13$

Finding the Equation from Two Points

Find the equation of the line passing through $(-1, 4)$ and $(3, -2)$.

Calculate the slope: $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-2 - 4}{3 - (-1)} = \frac{-6}{4} = -\frac{3}{2}$ = $m = -\frac{3}{2}$

Use point-slope form with one point: Using $(-1, 4)$: $y - 4 = -\frac{3}{2}(x - (-1))$ = $y - 4 = -\frac{3}{2}(x + 1)$

Simplify to slope-intercept form: $y - 4 = -\frac{3}{2}x - \frac{3}{2}$, so $y = -\frac{3}{2}x + \frac{5}{2}$ = $y = -\frac{3}{2}x + \frac{5}{2}$

Verify with the second point: Substitute $(3, -2)$: $-2 = -\frac{3}{2}(3) + \frac{5}{2} = -\frac{9}{2} + \frac{5}{2} = -\frac{4}{2} = -2$ ✓ = Verified!

Answer: $y = -\frac{3}{2}x + \frac{5}{2}$

Converting to Standard Form

Convert $y = \frac{2}{3}x - 4$ to standard form.

Eliminate the fraction: Multiply both sides by $3$: $3y = 2x - 12$ = $3y = 2x - 12$

Rearrange to standard form: Move $2x$ to the left: $-2x + 3y = -12$ = $-2x + 3y = -12$

Make the coefficient of $x$ positive: Multiply by $-1$: $2x - 3y = 12$ = $2x - 3y = 12$

Answer: $2x - 3y = 12$

Mistake: Confusing slope and y-intercept in $y = mx + b$

Why: Students sometimes think the first number is the y-intercept. In $y = 3x + 5$, the slope is $3$ (coefficient of $x$) and the y-intercept is $5$ (constant term).

Correct: Remember: $m$ comes before $b$ in the formula, and $m$ is multiplied by $x$. The y-intercept is the standalone number.

Mistake: Incorrect slope calculation: $\frac{x_2 - x_1}{y_2 - y_1}$

Why: Students flip the formula, putting $x$ on top instead of $y$.

Correct: Slope = rise over run = $\frac{y_2 - y_1}{x_2 - x_1}$. The $y$ values (vertical change) go on top.

Mistake: Sign errors in point-slope form

Why: When the point has a negative coordinate like $(-3, 2)$, students write $y - 2 = m(x - -3)$ incorrectly as $y - 2 = m(x - 3)$.

Correct: With point $(-3, 2)$: $x - (-3) = x + 3$, so the equation is $y - 2 = m(x + 3)$.

Mistake: Forgetting to distribute in point-slope form

Why: When expanding $y - 5 = 2(x - 3)$, students write $y - 5 = 2x - 3$ instead of $y - 5 = 2x - 6$.

Correct: Distribute the slope to BOTH terms inside the parentheses: $2(x - 3) = 2x - 6$.

Cell Phone Plans

Phone plans often have a fixed monthly fee plus a cost per minute or gigabyte used.

A plan costs 20 dollars per month plus 0.05 dollars per text message. The equation is $C = 0.05t + 20$, where $C$ is the total cost and $t$ is the number of texts.

Temperature Conversion

The relationship between Celsius and Fahrenheit is linear.

The formula $F = \frac{9}{5}C + 32$ converts Celsius to Fahrenheit. The slope $\frac{9}{5}$ means each degree Celsius equals $1.8$ degrees Fahrenheit.

Taxi Fare

Taxi companies typically charge a base fare plus a rate per kilometer.

If a taxi charges 3 euros to start and 1.50 euros per kilometer, the fare equation is $F = 1.5d + 3$, where $d$ is distance in kilometers.

Q: Which form should I use?

A: Use slope-intercept form when you need to graph quickly or identify slope/y-intercept. Use point-slope form when given a point and slope. Use standard form when working with systems of equations or when integer coefficients are required.

Q: How do I know if two equations represent the same line?

A: Convert both equations to the same form (usually slope-intercept). If they have the same slope and y-intercept, they represent the same line.

Q: What if the slope is zero or undefined?

A: If slope is zero, the line is horizontal: $y = b$. If slope is undefined (vertical line), the equation is $x = a$, which cannot be written in slope-intercept form.