Lesson 2: Graphs and Equations

New Concept

This lesson reveals the link between algebraic equations and their visual graphs. You'll learn that a graph is a picture of all possible solutions, allowing you to solve equations and inequalities like $Ax + By = C$ graphically.

What’s next

Next, you’ll work through interactive examples of solving equations graphically, followed by a sequence of practice cards to master this new skill.

Property

A solution of an equation is a value of the variable that makes the equation true.

Examples

$x = -2$ is a solution of the equation $4x^3 + 5x^2 - 8x = 4$ because substituting $-2$ for $x$ results in $4(-8) + 5(4) - 8(-2) = -32 + 20 + 16 = 4$.

To solve $3(2x - 5) - 4x = 2x - (6 - 3x)$, simplify both sides to $2x - 15 = 5x - 6$. Isolating the variable gives $-3x = 9$, so the solution is $x = -3$.

Property

Although a linear equation can have at most one solution, a linear inequality can have many solutions.
If we multiply or divide both sides of an inequality by a negative number, we must reverse the direction of the inequality.

Examples

To solve $5 - 2x < 2$, first subtract 5 from both sides to get $-2x < -3$. When you divide by $-2$, you must reverse the inequality to get the solution $x > \frac{3}{2}$.

To solve the inequality $-2x > -8$, we divide both sides by $-2$. Because we are dividing by a negative number, we must reverse the inequality symbol, which gives the solution $x < 4$.

Property

An equation in two variables, such as $y = 4x - 2$, has many solutions. Each solution consists of an ordered pair of values, one for $x$ and one for $y$, that together satisfy the equation (make the equation true.)

Examples

The ordered pair $(1, 3)$ is not a solution of $y = 4x - 2$ because substituting the values gives $4(1) - 2 = 2$. Since the y-value is 3, and $3 \ne 2$, the pair is not a solution.

Property

An equation in two variables, such as $-3x + 4y = 24$, has many solutions. Each solution consists of an ordered pair of values, one for $x$ and one for $y$, that together satisfy the equation (make the equation true.)

Examples

The ordered pair $(4, 3)$ is not a solution of $-3x + 4y = 24$ because substituting the values gives $-3(4) + 4(3) = -12 + 12 = 0$, which does not equal 24.

Property

The graph of an equation in two variables is just a picture of all its solutions. If a point lies on the graph of an equation, it is a solution of the equation.

Examples

If a point $(p, q)$ does not lie on the graph of $Ax + By = C$, it means that the ordered pair $(p, q)$ is not a solution, and the statement $Ap + Bq = C$ is not true.

Property

We can use graphs to find solutions to equations in one variable.

Examples

To solve the equation $150 = 285 - 15x$ using the graph of $y = 285 - 15x$, find the point on the graph where the y-coordinate is 150. The x-coordinate of that point is $x=9$, which is the solution.

To solve the inequality $285 - 15x \ge 150$ using the graph of $y = 285 - 15x$, find all points where the y-coordinate is 150 or more. The x-coordinates for these points are all values less than or equal to 9, so $x \le 9$.

Property

We can use graphs to find solutions to equations and inequalities in one variable.

Examples

Using the graph of $y = 3x - 2$:

• To solve the equation $10 = 3x - 2$: Find the point on the line where the y-coordinate is 10. The corresponding x-coordinate, $x=4$, is the solution.