Review: SAT heart of algebra linear equations

What is a linear equation?

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

Identify the slope in $\displaystyle y = 3x + 2$.

The slope is 3, indicating the line rises 3 units for every 1 unit it moves right.

True or false: The equation $\displaystyle y = 0.5x + 4$ is linear.

True, because it represents a straight line with a constant slope.

Fill in the blank: The y-intercept of $\displaystyle y = -2x + 5$ is _____

5, the value where the line crosses the y-axis.

Difference between slope and y-intercept.

Slope indicates the line's steepness, while the y-intercept is where the line crosses the y-axis.

What does a slope of 0 indicate?

The line is horizontal, indicating no change in y as x changes.

How to find the x-intercept of $\displaystyle 2x + 3y = 6$?

Set $\displaystyle y = 0$ and solve for $\displaystyle x$: $\displaystyle 2x = 6$, so $\displaystyle x = 3$.

What does it mean if two lines are parallel?

They have the same slope and will never intersect.

True or false: $\displaystyle y = -x + 1$ and $\displaystyle y = x + 1$ are parallel.

False, because they have different slopes (-1 and 1).

What is the solution to the system $\displaystyle y = 2x + 1$ and $\displaystyle y = -x + 5$?

Set the equations equal: $\displaystyle 2x + 1 = -x + 5$, leading to $\displaystyle x = \frac{4}{3}$, then find $\displaystyle y$.

Identify the slope in the equation $\displaystyle -3y + 6x = 9$.

Rewrite to slope-intercept form: $\displaystyle y = 2x - 3$. The slope is 2.

What is a system of equations?

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

True or false: The point (0, 0) is a solution to $\displaystyle y = 3x$.

True, because it satisfies the equation.

How do you graph the line $\displaystyle y = 4$?

Draw a horizontal line across the y-axis at y = 4.

What does it mean if two lines intersect?

They have different slopes, and their intersection point is the solution to the system.

Fill in the blank: A vertical line has an undefined _____

slope, meaning it rises infinitely without moving left or right.

True or false: $\displaystyle y = 2x^2 + 3$ is a linear equation.

False, because it contains a variable squared, making it quadratic.