Lesson 4: Equations and Graphs

New Concept

This lesson bridges algebra and geometry. You'll master solving equations with negative numbers and visualize these relationships by plotting points and graphing straight lines on the full, four-quadrant Cartesian coordinate system.

What’s next

You'll start with practice cards for solving equations, then move to interactive examples for plotting and graphing.

Property

We solve an equation by undoing in reverse order the operations performed on the variable. We can think of any string of terms as a sum, where the $+$ and $-$ symbols tell us the sign of the term that follows. To solve, first add or subtract terms to isolate the variable term, then multiply or divide to find the variable's value.

Examples

• To solve $12 - 4x = -8$, first subtract 12 from both sides to get $-4x = -20$. Then divide by $-4$ to find $x = 5$.
• To solve $-3y + 7 = 22$, first subtract 7 from both sides to get $-3y = 15$. Then divide by $-3$ to get $y = -5$.
• To solve $\frac{x-5}{3} = -2$, first multiply both sides by 3 to get $x-5 = -6$. Then add 5 to both sides to find $x = -1$.

Explanation

Solving an equation is like being a detective to find the variable's hidden value. You reverse the equation's steps, undoing each operation one by one until the variable is alone on one side of the equals sign.

Property

To display the values of two variables, we use two number lines. The horizontal number line is called the $x$-axis, and the vertical number line is the $y$-axis. The point where the two axes intersect is called the origin. The two axes divide the plane into four regions called quadrants, numbered 1 through 4 counter-clockwise around the origin.

Examples

• Points in Quadrant I, like $(4, 6)$, have a positive x-coordinate (a move to the right) and a positive y-coordinate (a move up).
• Points in Quadrant III, like $(-1, -5)$, have a negative x-coordinate (a move to the left) and a negative y-coordinate (a move down).
• The origin is the starting point $(0, 0)$ where the x-axis and y-axis meet.

Explanation

Think of the Cartesian coordinate system as a map for numbers. It uses two perpendicular lines, the x-axis (horizontal) and y-axis (vertical), to give a unique address to any point on a flat surface, letting us visualize equations.

Property

The location of a point in the plane is given by an ordered pair $(a, b)$, where $a$ is the $x$-coordinate and $b$ is the $y$-coordinate. Starting at the origin, move along the $x$-axis to the right if the $x$-coordinate is positive, and to the left if it is negative. From there, move up if the $y$-coordinate is positive, and down if it is negative.

Examples

• To plot the point $(3, 4)$, you start at the origin, move 3 units to the right along the x-axis, and then move 4 units up.
• To plot the point $(-5, -2)$, you start at the origin, move 5 units to the left along the x-axis, and then move 2 units down.
• A point on an axis, like $(0, 6)$, has an x-coordinate of 0. You do not move left or right, only 6 units up from the origin.

Explanation

Plotting an ordered pair $(x, y)$ is like a two-part instruction. The x-coordinate is your 'run' (horizontal move from the origin), and the y-coordinate is your 'jump' (vertical move). Together, they pinpoint the exact location on the graph.

Property

An equation of the form $y = ax + b$, where $a$ and $b$ are constants, is a linear equation. Its graph is a straight line. To graph an equation:

1. Make a table of values by choosing input values for $x$ and calculating the output values for $y$.
2. Choose appropriate scales and label the axes.
3. Plot the points and connect them with a straight line.

Examples

• To graph $y = x + 4$, find points like $(-1, 3)$, $(0, 4)$, and $(2, 6)$. Plotting these points and connecting them creates the line.
• For the equation $y = -3x + 2$, you can use points like $(0, 2)$, $(1, -1)$, and $(2, -4)$ to draw its graph.
• To graph $y = \frac{1}{2}x - 3$, choose multiples of 2 for $x$. Points like $(-2, -4)$, $(0, -3)$, and $(4, -1)$ will lie on the line.

Explanation

Graphing a linear equation turns algebra into a picture. Every point on the line is a solution to the equation. By finding just a few solution points and connecting them, you can draw the line that represents all possible solutions.