Writing Linear Equations from Context

Identifying Slope and Y-Intercept in Context

In a real-world linear model:

• Slope ($m$) = rate of change (how much $y$ changes per unit of $x$)
• Y-intercept ($b$) = initial/starting value (the value of $y$ when $x = 0$)

Example: "A plumber charges $50 for the service call plus $30 per hour."

• $x$ = number of hours
• $y$ = total charge
• $m = 30$ ($30 per hour)
• $b = 50$ ($50 starting fee)
• Equation: $y = 30x + 50$

Writing Equations from Word Problems

Step-by-step process:

1. Define variables. What does $x$ represent? What does $y$ represent?
2. Find the rate of change (slope) — look for "per", "each", "every", "for each additional".
3. Find the starting value (y-intercept) — look for "initial", "base", "fixed", "already".
4. Write the equation in $y = mx + b$ form.

Writing Equations from Tables

Given a table of values:

1. Calculate the slope: $m = \frac{\Delta y}{\Delta x}$ using any two rows.
2. Find the y-intercept: substitute a known point into $y = mx + b$ and solve for $b$.

$x$	$y$
2	11
5	20
8	29

Slope: $m = \frac{20 - 11}{5 - 2} = \frac{9}{3} = 3$

Using $(2, 11)$: $11 = 3(2) + b$ → $b = 5$

Equation: $y = 3x + 5$

Writing Equations from Graphs

1. Read the y-intercept from the graph.
2. Pick two clear grid-intersection points and calculate the slope.
3. Write $y = mx + b$.

Interpreting Slope and Intercept

The SAT often asks you to interpret what the slope or y-intercept means in context.

Example: "The equation $C = 0.15m + 35$ models the monthly cost $C$ (in dollars) of a phone plan where $m$ is the number of minutes used."

• The 0.15 means each additional minute costs $0.15.
• The 35 means the base monthly charge is $35 (even with 0 minutes).

Comparing Linear Models

Some SAT questions present two scenarios and ask when they're equal or which is better.

Example: Plan A costs $20 plus $3/mile. Plan B costs $50 plus $1/mile.

• Plan A: $y = 3x + 20$
• Plan B: $y = x + 50$
• Equal when $3x + 20 = x + 50$ → $2x = 30$ → $x = 15$ miles

Tip 1: Look for Keywords

• "Per", "each", "rate", "every" → slope
• "Initial", "starting", "base", "flat fee", "already" → y-intercept
• "Total", "altogether" → the output variable $y$

Tip 2: Plug In a Known Value to Check

After writing your equation, plug in a known data point to verify it works.

Tip 3: Units Guide Your Setup

The slope's units are "y-units per x-unit" (e.g., dollars per hour, miles per minute). This helps you set up the equation correctly.

Tip 4: Read What the Question Asks

Some questions ask for the equation, some for an interpretation, and some for a prediction. Don't over-solve.

Tip 5: Use the Desmos Calculator

On the Digital SAT, you can enter data into Desmos and have it compute the line of best fit. Useful for table-based problems.