Solving linear equations word problems for a test

What is a linear equation?

An equation that forms a straight line when graphed, typically in the form $\displaystyle y = mx + b$.

Example of a linear equation in a word problem.

If a shirt costs $\displaystyle 20 and you buy x shirts, the total cost is$20x.

True or false: Linear equations can have curves.

False, because linear equations represent straight lines.

Difference between dependent and independent variables.

Independent variables are inputs ($\displaystyle x$), while dependent variables are outputs ($\displaystyle y$) that depend on $\displaystyle x$.

Fill in the blank: $\displaystyle y = 2x + 3$ is an example of a ___ equation.

linear

How to set up a linear equation from a problem?

Identify the variables, the relationship between them, and express it in equation form.

What does slope represent in a linear equation?

It represents the rate of change of $\displaystyle y$ with respect to $\displaystyle x$.

Solve: You buy 3 notebooks for $\displaystyle 2 each, plus$5 for a pen. Total cost?

$\displaystyle 2(3) + 5 = 11$ dollars.

True or false: The graph of $\displaystyle y = 5$ is a line.

True, because it represents a horizontal line at $\displaystyle y = 5$.

Example of a real-life scenario for linear equations.

Calculating total rent when $\displaystyle x$ roommates pay $400 each.

What does the y-intercept represent?

The value of $\displaystyle y$ when $\displaystyle x = 0$ in a linear equation.

Solve: If you earn $15 per hour and work x hours, your pay is.

$\displaystyle 15x$ dollars.

Difference between one-variable and two-variable equations.

One-variable has one unknown, e.g., $\displaystyle x + 5 = 10$. Two-variable has two, e.g., $\displaystyle y = 2x + 1$.

Fill in the blank: The equation $\displaystyle y = mx + b$ has slope m and ___ b.

y-intercept

What is the first step in solving a word problem?

Identify the unknowns and assign variables to them.

True or false: The solution to a linear equation is the point where two lines intersect.

True, because that point satisfies both equations.

Solve: You have $\displaystyle 50, and you spend$x$ on a movie ticket. What's left?

$\displaystyle 50 - x$ dollars.

What is a system of linear equations?

A set of two or more linear equations with the same variables.

Example of a linear equation for distance.

Distance = rate × time: $\displaystyle d = rt$.

Fill in the blank: To find intersection points, solve ___ equations.

simultaneous