Problem 63 - College Algebra with Modeling and Visualization

Question

Determine the \(x\) - and \(y\) -intercepts on the graph of the equation. Graph the equation. $$ y-3 x=7 $$

Step-by-Step Solution

Answer

x-intercept: \( -\frac{7}{3} \), y-intercept: \( 7 \). Graph is a straight line through these points.

Step 1: Finding the x-intercept

To find the x-intercept, we set \( y = 0 \) and solve for \( x \). The equation is \( y - 3x = 7 \). Setting \( y \) to zero gives us \( 0 - 3x = 7 \). Solve for \( x \): \[ -3x = 7 \]\[ x = -\frac{7}{3} \]. The x-intercept is at \( \left( -\frac{7}{3}, 0 \right) \).

Step 2: Finding the y-intercept

To find the y-intercept, we set \( x = 0 \) and solve for \( y \). The equation is \( y - 3x = 7 \). Setting \( x \) to zero gives us \( y - 0 = 7 \), so \( y = 7 \). The y-intercept is at \( (0, 7) \).

Step 3: Graphing the Equation

Using the intercepts found in Steps 1 and 2, we can graph the equation. Plot the x-intercept \( \left( -\frac{7}{3}, 0 \right) \) and the y-intercept \( (0, 7) \) on a coordinate plane. Draw a straight line through these two points, as the equation represents a linear function.

Key Concepts

Intercepts

Intercepts are the points where a graph crosses the axes. There are two main types: **x-intercepts** and **y-intercepts**. - An **x-intercept** occurs where the graph crosses the x-axis. At this point, the value of y is always zero. To find the x-intercept, substitute zero for y in the equation and solve for x. In our exercise, we found the x-intercept of the equation \( y - 3x = 7 \) by setting \( y = 0 \). This gave us \( x = -\frac{7}{3} \). Thus, the x-intercept is \( \left( -\frac{7}{3}, 0 \right) \). - A **y-intercept** is where the graph crosses the y-axis, and the value of x is zero. Finding the y-intercept involves substituting zero for x in the equation and solving for y. From the example, by setting \( x = 0 \), we calculated that \( y = 7 \). Therefore, the y-intercept is \( (0, 7) \). Intercepts are crucial because they simplify the process of drawing graphs and understanding the function's behavior at the axes.

Graphing

Graphing a linear equation involves plotting points and drawing a line through them. It's a visual way to represent equations. Here's how you can do it effectively: - Start by finding the intercepts, as demonstrated earlier. These points will be your anchor points on the graph. For the equation \( y - 3x = 7 \), we already found the intercepts to be \( \left( -\frac{7}{3}, 0 \right) \) and \( (0, 7) \). - Next, plot these intercepts on a coordinate plane. The x-intercept \( \left( -\frac{7}{3}, 0 \right) \) will be on the x-axis, while the y-intercept \( (0, 7) \) will be on the y-axis. - After plotting, draw a straight line through the points. Linear equations like this one form a straight line because they represent a constant rate of change. This line is the graph of the equation, and any point on it satisfies the equation.

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. In solving linear equations, algebraic techniques help isolate variables and solve for unknowns. Let's take a closer look: - Consider the equation \( y - 3x = 7 \). Algebraic manipulation begins by deciding which variable to isolate. For intercepts, set one variable to zero: - For the **x-intercept**, set \( y = 0 \) and solve the equation \( 0 - 3x = 7 \). This leads to \( -3x = 7 \) and then \( x = -\frac{7}{3} \). - For the **y-intercept**, set \( x = 0 \) and solve \( y - 0 = 7 \), simplifying directly to \( y = 7 \). - Understanding the concepts of variables and coefficients is essential here. Coefficients (numbers multiplying the variables) play a significant role in shaping the graph and determining the direction and steepness of the line. Algebra allows us to manipulate equations to find specific solutions, like intercepts, which are key for graphing linear equations.