Problem 41 - Algebra and Trigonometry

Question

\(41-48\) Find the \(x\) - and \(y\) -intercepts of the graph of the equation. $$ y=x-3 $$

Step-by-Step Solution

Answer

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

Step 1: Finding the y-intercept

To find the y-intercept of the graph of the equation, set the value of \(x\) to 0 in the equation \(y = x - 3\). This will give us the value of \(y\) when the graph intersects the y-axis. Substituting \(x = 0\), the equation becomes \(y = 0 - 3\), so \(y = -3\). Thus, the y-intercept is the point (0, -3).

Step 2: Finding the x-intercept

To find the x-intercept of the graph of the equation, set the value of \(y\) to 0 in the equation \(y = x - 3\). This will give us the value of \(x\) when the graph intersects the x-axis. Substituting \(y = 0\), the equation becomes \(0 = x - 3\). Solving for \(x\), we get \(x = 3\). Thus, the x-intercept is the point (3, 0).

Key Concepts

Linear Equations

Linear equations are fundamental in algebra and help us describe a straight line on a coordinate system. A linear equation is typically in the form of \(y = mx + b\), where \(m\) represents the slope, and \(b\) is the y-intercept. The equation \(y = x - 3\) is a simple linear equation where the slope \(m = 1\) and the y-intercept \(b = -3\). The slope indicates how steep the line is; in this case, it's not too steep since the slope is 1.
Linear equations are helpful because they allow us to predict the relationship between variables. They represent constant change, meaning as one variable increases or decreases linearly, so does the other. This property makes linear equations widely applicable in real-life situations. For instance, if you are driving at a constant speed, the distance over time can be described as a linear equation.

• The graph of the equation is always a straight line.
• It confirms a consistent rate of change.
• Generally used to model simple relationships.

x-intercept

The x-intercept is a point where a graph crosses the x-axis. Here, the value of \(y\) is always zero. To find the x-intercept of a linear equation like \(y = x - 3\), we set \(y\) to 0 and solve for \(x\). Here’s how:
The equation becomes \(0 = x - 3\). Solving for \(x\) means isolating \(x\) on one side of the equation. In this case, we add 3 to both sides to find \(x = 3\). That means the graph crosses the x-axis at the point \((3, 0)\).
This is useful because the x-intercept can tell us how many units from the origin the graph crosses the horizontal axis, providing a point of reference on the graph.

• Setting \(y = 0\) helps locate the x-intercept.
• The x-intercept in this problem is \((3, 0)\).
• It gives insight into where the function nullifies or equals zero.

y-intercept

The y-intercept is where the graph intersects the y-axis. At this intersection, the value of \(x\) is always zero. For the equation \(y = x - 3\), we determine the y-intercept by setting \(x\) to 0. Substituting into the equation, we find \(y = 0 - 3\), hence \(y = -3\). Therefore, the graph crosses the y-axis at \((0, -3)\).
Understanding the y-intercept is critical because it represents the starting point of a function when \(x\) is 0. In our equation, it signifies the output value of \(-3\) when no input (or \(x\)) is present.
The y-intercept also tells us exactly where on the y-axis a graph will start its journey across the coordinate plane. This point provides insight into the initial value of a relation modeled by a linear equation. Some good points to note:

• Found by setting \(x = 0\).
• In our example, it results in the point \((0, -3)\).
• Essential for graphing a line quickly and understanding initial conditions.