Problem 17 - Algebra 1

Question

Find the \(x\) -intercept of the graph of the equation. $$ x+3 y=5 $$

Step-by-Step Solution

Answer

The x-intercept of the graph of the equation is at \(x = 5\).

Step 1: Set y = 0

To find the x-intercept, plug in 0 for \(y\) in the equation to get \(x + 3(0) = 5\).

Step 2: Solve for x

Simplify the equation from step 1 to isolate \(x\). That gives you \(x = 5\).

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 a single variable. Linear equations are fundamental in algebra and they form a straight line when graphed on a coordinate plane. They are usually written in the form of
\( ax + by = c \), where \(a\) and \(b\) are coefficients, \(x\) and \(y\) are variables, and \(c\) is a constant. The equation \(x + 3y = 5\) from our exercise is an example of a linear equation.

In the context of graphing, the equation represents a line and we are often interested in finding specific points, such as where the line crosses the axes. These points are known as intercepts. The x-intercept, which is the focus of our exercise, is the point where the line crosses the x-axis (where \(y=0\)).

Solving Equations

Solving equations is a core skill in algebra. The goal is to find the value(s) of the variable(s) that make the equation true. To solve linear equations like the one in our exercise, we typically perform operations that will isolate the variable of interest on one side of the equation.

Here's a step-by-step breakdown using the exercise example: We start by identifying we want to solve for the x-intercept, and we know that at the x-intercept, \(y=0\). So, we set \(y=0\) in the linear equation: \[{x + 3(0) = 5}\].
This simplifies to \[{x = 5}\] as \(3(0)\) is just \(0\). Thereby, we have isolated \(x\) on one side of the equation, and \(5\) is the value that makes this equation true. Thus, \(x=5\) is the solution, and it means that the x-intercept is at the point \(5,0\) on the coordinate graph.

Coordinate Graphing

Coordinate graphing is a method to visually represent equations on a plane using an ordered pair of numbers, known as coordinates. The plane is defined by a horizontal axis, typically the x-axis, and a vertical axis, the y-axis. Points are plotted on this plane with reference to these axes.

The point where the line crosses the x-axis, the x-intercept, has a y-coordinate of zero. Similarly, the y-intercept is where the line crosses the y-axis, resulting in an x-coordinate of zero. In our example, the x-intercept was found to be \(5,0\); if you were to plot this on a graph, you'd place a point where the x-axis is at \(5\), and since the y-coordinate is \(0\), the point would be exactly on the x-axis itself.