Solving Problems Involving Linear Functions

Slope or Rate of Change ( $m$ ): Indicated by words such as "per," "each," "constant rate," "increases by," or "decreases by."
Initial Value or $y$ -intercept ( $b$ ): Indicated by phrases like "starting at" or "initial fee."
Independent Variable ( $x$ ): Represents the input, usually measuring quantity or time (e.g., "number of items").
Dependent Variable ( $y$ or $f(x)$ ,): Represents the output or result (e.g., "total cost" or "how much").

Step 1: Read the problem to identify what needs to be solved.
Step 2: List given data, specifically identifying the independent variable ( $x$ ), slope ( $m$ ), $y$ -intercept ( $b$ ), and dependent variable ( $f(x)$ ).
Step 3: Formulate the linear function $f(x) = mx + b$ and solve for the unknown variable.
Step 4: Check the answer by substituting the values back into the function.

Direct Variation ( $f(x) = mx$ ):
- Mr. Han: Sells merchandise at $200$ per photocard. To find the quantity sold ( $x$ ) for an earning of $2600$ pesos: $f(x) = 200x$ $2600 = 200x$ $x = 13$
- Lee: Sells charms for $25$ pesos each. If selling $40$ pieces daily for $3$ days ( $x = 120$ ): $f(x) = 25x$ $f(120) = 25 \times 120$ $f(120) = 3000$
Slope-Intercept Form ( $f(x) = mx + b$ ):
- Momo: Has initial savings ( $b$ ) of $150$ pesos and saves $10$ pesos weekly ( $m$ ). Total savings after $5$ weeks: $f(x) = 10x + 150$ $f(5) = 10 \times 5 + 150$ $f(5) = 200$
- Conclusion: Momo reaches the required $200$ pesos for a music subscription after $5$ weeks.