Problem 22 - Algebra 1

Question

Find the \(x\) -intercept of the graph of the equation. $$ -x+3 y=27 $$

Step-by-Step Solution

Answer

The x-intercept of the graph of the given equation is \( x = -27 \).

Step 1: Set up the equation

Rewrite the provided equation \( -x + 3y = 27 \) as it is.

Step 2: Introduce y as zero

At the x-intercept, the value of y is zero. So replace y with 0 in the equation. The equation should now look like this: \( -x + 3*0 = 27 \).

Step 3: Solve for x

Solving for x gives \( x = -27 \).

Key Concepts

x-intercept

An x-intercept is a point where a graph crosses the x-axis. This is a fundamental concept in understanding linear equations. On the x-axis, the value of y is always zero. Therefore, in any equation involving x and y, to find the x-intercept, you simply need to set y to zero.

Consider how it is applied: in the equation \( -x + 3y = 27 \), we determine the x-intercept by letting \( y = 0 \). As a result, the equation simplifies to \( -x + 3 \times 0 = 27 \).

This simplifies further to \( -x = 27 \), thus giving \( x = -27 \). The x-intercept is therefore at the point \((-27, 0)\). Remember, the x-intercept indicates where the graph of the equation crosses the x-axis, which is a critical aspect of graph interpretation.

Linear Equations

Linear equations are mathematical statements that show a straight-line relationship between two variables. They can be written in various forms, one of which is the standard form: \( Ax + By = C \). In these equations, both x and y are raised only to the first power, meaning they create a line when graphed.

For instance, \(-x + 3y = 27\) is a linear equation. Here, the coefficients of x and y carry vital information about the slope and intercepts of the line. The structure of linear equations allows for straightforward plotting on a graph.

Understanding the elements of linear equations includes:

• Intercepts: Points where the line crosses the axes, such as x- and y-intercepts.
• Slope: A measure of how steep the line is. In the equation's slope-intercept form, \(y = mx + b\), \(m\) represents the slope.

All these elements play crucial roles in graphing and interpreting the behavior of linear relationships.

Solving Equations

Solving equations involves finding the value of the variable that makes the equation true. This could be done by simplifying the equation step by step until the variable is isolated.

For example, solving the equation \(-x + 3y = 27\) for \(x\), when \(y = 0\), involves replacing y with zero which leads to \(-x + 0 = 27\).

Next, we solve for \(x\) by isolating it on one side. This requires a fundamental understanding of operations like addition, subtraction, and the handling of negatives. Here, \(-x = 27\) becomes \(x = -27\) as we solve for \(x\).

Key strategies when solving equations include:

• Performing the same operations on both sides of the equation to maintain equality.
• Systematically isolating the variable by reversing operations such as addition or subtraction.

Mastering these techniques is crucial for effectively solving various types of algebraic equations.