Writing an Equation from a Point and Slope

Write the equation of a line with slope $m = 3$ that passes through the point $(2, 5)$.

Identify the values: Slope $m = 3$, Point $(x_1, y_1) = (2, 5)$ = $m = 3$, $x_1 = 2$, $y_1 = 5$

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

Substitute values: $y - 5 = 3(x - 2)$ = $y - 5 = 3(x - 2)$

Answer: $y - 5 = 3(x - 2)$

Converting to Slope-Intercept Form

Convert $y - 5 = 3(x - 2)$ to slope-intercept form.

Distribute on the right side: $y - 5 = 3x - 6$ = $y - 5 = 3x - 6$

Add 5 to both sides: $y = 3x - 6 + 5$ = $y = 3x - 1$

Verify the form: Slope $m = 3$, y-intercept $b = -1$ = $y = 3x - 1$

Answer: $y = 3x - 1$

Finding an Equation from Two Points

Find the equation of the line passing through $(1, 4)$ and $(3, 10)$.

Calculate the slope: $m = \frac{10 - 4}{3 - 1} = \frac{6}{2} = 3$ = $m = 3$

Choose one point: Using $(1, 4)$ as $(x_1, y_1)$ = $x_1 = 1$, $y_1 = 4$

Write point-slope form: $y - 4 = 3(x - 1)$ = $y - 4 = 3(x - 1)$

Verify with second point: $10 - 4 = 3(3 - 1)$ gives $6 = 6$ \checkmark = Equation verified!

Answer: $y - 4 = 3(x - 1)$ or equivalently $y = 3x + 1$

Writing a Parallel Line Equation

Write the equation of a line parallel to $y = 2x + 5$ that passes through $(-1, 3)$.

Identify the slope: Parallel lines have the same slope. From $y = 2x + 5$, $m = 2$ = $m = 2$

Use the given point: Point: $(-1, 3)$, so $x_1 = -1$, $y_1 = 3$ = Point identified

Write point-slope form: $y - 3 = 2(x - (-1))$ = $y - 3 = 2(x + 1)$

Simplify if needed: $y - 3 = 2x + 2$, so $y = 2x + 5$ = Wait - same as original?

Check calculation: $y = 2x + 2 + 3 = 2x + 5$... but this passes through $(-1, 3)$! = $y - 3 = 2(x + 1)$ is correct

Answer: $y - 3 = 2(x + 1)$ or $y = 2x + 5$

Writing a Perpendicular Line Equation

Write the equation of a line perpendicular to $y = \frac{1}{2}x - 3$ that passes through $(4, 1)$.

Find the original slope: From $y = \frac{1}{2}x - 3$, the slope is $m = \frac{1}{2}$ = $m_{original} = \frac{1}{2}$

Find perpendicular slope: Perpendicular slope = negative reciprocal: $m_{\perp} = -\frac{1}{\frac{1}{2}} = -2$ = $m_{\perp} = -2$

Use point-slope form: $y - 1 = -2(x - 4)$ = $y - 1 = -2(x - 4)$

Convert to slope-intercept: $y - 1 = -2x + 8$, so $y = -2x + 9$ = $y = -2x + 9$

Answer: $y - 1 = -2(x - 4)$ or $y = -2x + 9$

Mistake: Forgetting to change the sign when $x_1$ or $y_1$ is negative

Why: In $y - y_1 = m(x - x_1)$, if $y_1 = -3$, then $y - (-3) = y + 3$, not $y - 3$.

Correct: Always write the subtraction first, then simplify: $y - (-3) = y + 3$

Mistake: Using the wrong point for $(x_1, y_1)$

Why: When given two points, students sometimes mix coordinates from different points.

Correct: Pick ONE point completely. Both coordinates must come from the same point.

Mistake: Distributing incorrectly

Why: In $y - 5 = 3(x - 2)$, students may forget to distribute to both terms.

Correct: $3(x - 2) = 3x - 6$, not $3x - 2$

Mistake: Confusing parallel and perpendicular slopes

Why: Parallel lines have the SAME slope. Perpendicular slopes are NEGATIVE RECIPROCALS.

Correct: Parallel: same $m$. Perpendicular: if $m = \frac{2}{3}$, then $m_{\perp} = -\frac{3}{2}$

Business Growth Projection

Companies use linear models to project growth when they know their current position and rate of change.

A startup has 500 users today and gains 50 new users per day. The equation $y - 500 = 50(x - 0)$ models their user count, where $x$ is days from today.

Temperature Change

Scientists model temperature changes using linear equations when the rate is constant.

If the temperature at 2 PM was 18°C and drops 3°C per hour, the equation $T - 18 = -3(t - 2)$ models the temperature, where $t$ is the hour (in PM).

Q: When should I use point-slope form instead of slope-intercept form?

A: Use point-slope form when you know a point and the slope but don't know the y-intercept. It's quicker than finding $b$ first.

Q: Does it matter which point I choose when given two points?

A: No! Using either point will give you an equivalent equation. The final line is the same - just the form looks different until you simplify.

Q: How do I know if two lines are parallel?

A: Two lines are parallel if and only if they have the same slope but different y-intercepts. In point-slope form, the $m$ values must be equal.