Problem 43 - College Algebra with Corequisite Support

Question

For the following exercises, find the \(x\) - and \(y\) -intercepts of each equation. $$ h(x)=3 x-5 $$

Step-by-Step Solution

Answer

y-intercept: (0, -5); x-intercept: \(\left(\frac{5}{3}, 0\right)\).

Step 1: Understand the Equation

The equation given is in the form of a linear function, \( h(x)=3x-5 \). To find the intercepts, we'll consider the equation as \( y = 3x - 5 \).

Step 2: Find the y-intercept

To find the y-intercept, set \(x = 0\) in the equation and solve for \(y\). Thus, \( y = 3(0) - 5 = -5 \). So the y-intercept is at the point \((0, -5)\).

Step 3: Find the x-intercept

To find the x-intercept, set \(y = 0\) in the equation and solve for \(x\). So, \( 0 = 3x - 5 \). Solving for \(x\), we get \(3x = 5\), therefore \( x = \frac{5}{3} \). Thus, the x-intercept is at the point \(\left(\frac{5}{3}, 0\right)\).

Step 4: Review the intercepts

We have calculated the intercepts as follows: the y-intercept is \((0, -5)\) and the x-intercept is \(\left(\frac{5}{3}, 0\right)\). Double-check these values by plugging them back into the original equation to ensure they satisfy the equation.

Key Concepts

Understanding x-intercept

The x-intercept is the point where the graph of a function crosses the x-axis. At this point, the value of the function is zero. In other words, the y-value is 0. To calculate the x-intercept, you set the function equal to zero and solve the resulting equation for the x-variable.
For example, in the equation \[ y = 3x - 5 \] we find the x-intercept by replacing y with 0 resulting in the equation \[ 0 = 3x - 5 \]. This requires solving for x:

• Add 5 to both sides to isolate the term with x: \[ 3x = 5 \]
• Divide both sides by 3 to solve for x: \[ x = \frac{5}{3} \]

This means the graph crosses the x-axis at the point \( \left(\frac{5}{3}, 0\right) \). Understanding how to find the x-intercept helps in graphing linear equations accurately.

Exploring y-intercept

The y-intercept is where the graph crosses the y-axis. At this intercept, the x-value is 0. This makes finding the y-intercept a straightforward task. Simply set x to 0 in the linear equation and solve for y.
For our equation \[ y = 3x - 5 \], setting x to 0 gives:

• Substitute 0 for x: \[ y = 3(0) - 5 \]
• Simplifying this gives: \[ y = -5 \]

This means the graph crosses the y-axis at the point \( (0, -5) \). Finding the y-intercept is essential for sketching the graph of a line, as it provides one point through which the graph passes.

Demystifying algebraic calculations in linear equations

Algebraic calculations are the mathematical operations we use to manipulate equations and find solutions. In the context of linear equations, such as \[ y = 3x - 5 \], these calculations include solving for intercepts and rearranging terms. These operations involve basic algebraic techniques that are essential for both solving equations and understanding their graphs.
For example:

• To find intercepts, set the respective variable to zero and solve the equation.
• Use operations such as addition, subtraction, multiplication, and division.

These calculations allow you to explore various characteristics and solutions of the equation. They help in understanding how changes to one part of the equation affect the graph as a whole.