Problem 63 - Algebra 1

Question

Find the x-intercept and the y-intercept of the graph of the equation. $$y=\frac{1}{3} x-\frac{7}{3}$$

Step-by-Step Solution

Answer

The y-intercept is at (0, -7/3) and the x-intercept is at (7, 0).

Step 1: Find the Y-Intercept

Set x = 0 in the equation \(y=\frac{1}{3} x-\frac{7}{3}\) and solve for y. This will give the y-coordinate of the y-intercept.

Step 2: Substitute X=0 in the Equation

The equation becomes \(y=\frac{1}{3} (0)-\frac{7}{3} = -\frac{7}{3}\). Therefore, the y-intercept is at (0, -7/3).

Step 3: Find the X-Intercept

Set y = 0 in the equation \(y=\frac{1}{3} x-\frac{7}{3}\) and solve for x. This will give the x-coordinate of the x-intercept.

Step 4: Substitute Y=0 in the Equation

The equation becomes \(0=\frac{1}{3} x-\frac{7}{3}\). Solving for x, x equals 7. Therefore, the x-intercept is at (7, 0).

Key Concepts

Linear Equations

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and the first power of a single variable. Linear equations form straight lines when graphed on a coordinate plane and have at most one variable raised to the power of one. The general formula for a linear equation in two variables, x and y, is given by \(y = mx + b\), where \(m\) represents the slope of the line and \(b\) indicates the y-intercept.

Understanding linear equations is crucial for finding the x-intercepts and y-intercepts which are points where the graph crosses the axes. To find the y-intercept, we set \(x = 0\) and solve for \(y\); to find the x-intercept, we set \(y = 0\) and solve for \(x\). For instance, using the equation \(y=\frac{1}{3} x-\frac{7}{3}\), we can find the intercepts by substituting the respective values for \(x\) and \(y\).

Graphing Linear Equations

Graphing linear equations involves plotting the relationship between two variables on a coordinate plane. The graph of any linear equation will result in a straight line. To graph a linear equation like \(y=\frac{1}{3} x-\frac{7}{3}\), we typically need at least two points.

The most straightforward points to use are the intercepts. The y-intercept is found by setting \(x = 0\), which is where the line crosses the y-axis. Conversely, the x-intercept is where \(y = 0\) and represents where the line crosses the x-axis. Plotting these two points and drawing a line through them will accurately represent the equation on a graph. By connecting the intercepts (0, -\frac{7}{3}) and (7, 0), the graph of the equation is formed.

Visualizing the Solution

When you plot these points and draw the line, you should observe that the slope \(\frac{1}{3}\) dictates how steep the line is, and the y-intercept \(\frac{7}{3}\) suggests the starting point on the y-axis. The slope is positive, meaning the line will slant upwards from left to right.

Algebraic Methods

Algebraic methods are systematic strategies used to solve equations and can be applied to find the intercepts of a linear equation. These methods include substitution and rearranging the equation to isolate a variable.

For our given equation \(y=\frac{1}{3} x-\frac{7}{3}\), to find the y-intercept, we perform substitution by replacing \(x\) with 0 and solve for \(y\). This gives us the point (0, -\frac{7}{3}). Similarly, to find the x-intercept, we substitute \(y\) with 0 and solve for \(x\), resulting in the point (7, 0).

Simplifying the Process

Understanding algebraic methods is essential as they not only allow us to determine specific points like intercepts but also empower us to manipulate and solve for variables in more complex equations. With practice, algebraic manipulation becomes an invaluable skill in graph analysis and helps in visualizing linear relationships.