Setting up a Linear Equation
Model real-world problems by identifying known quantities, defining variables, and translating word problems into linear equations in Grade 9 Algebra.
Key Concepts
Property To set up or model a linear equation to fit a real world application, we must first determine the known quantities and define the unknown quantity as a variable. Then, we begin to interpret the words as mathematical expressions using mathematical symbols. For example, a variable cost can be written as $0.10x$, while a fixed cost is a constant added or subtracted, such as in $C = 0.10x + 50$.
To model a linear equation: 1. Identify known quantities. 2. Assign a variable to represent the unknown quantity. 3. If there is more than one unknown quantity, find a way to write the second unknown in terms of the first. 4. Write an equation interpreting the words as mathematical operations. 5. Solve the equation. Be sure the solution can be explained in words, including the units of measure.
Examples One number is 10 more than another, and their sum is 52. Let the first number be $x$. The second is $x+10$. The equation is $x + (x+10) = 52$, so $2x=42$, and $x=21$. The numbers are 21 and 31. A taxi charges 3 dollars plus 2 dollars per mile. The total cost $C$ for a ride of $x$ miles is modeled by the equation $C = 2x + 3$. A 5 mile ride costs $C = 2(5) + 3 = 13$ dollars. Two streaming services have different plans. Plan A is 15 dollars a month. Plan B is 5 dollars a month plus 2 dollars per movie. To find when they cost the same for $m$ movies, set $15 = 5 + 2m$. Solving gives $10 = 2m$, so $m=5$ movies.
Common Questions
How do you set up a linear equation from a word problem?
First identify all known quantities and what you need to find. Define the unknown as a variable (usually x). Translate key phrases into math operations: 'more than' means add, 'times' means multiply, 'total' means equals. Then write the equation and solve.
What is a variable cost in a linear equation model?
A variable cost changes in proportion to some quantity, like the number of units produced. In a linear equation, the variable cost appears as the mx term, where m is the cost per unit and x is the number of units. A fixed cost appears as the constant b.
What is the difference between fixed and variable quantities in a linear model?
Fixed quantities stay constant regardless of the independent variable — like a one-time fee. Variable quantities change in proportion to the input — like cost per item. Together they form the equation Total = (variable rate × quantity) + fixed cost, which is the linear model f(x) = mx + b.