Problem 32 - Algebra 2

Question

Find the \(x\)-intercept and the \(y\)-intercept of the graph of each equation. Then graph the equation. \(y=x\)

Step-by-Step Solution

Answer

Both intercepts are at (0, 0), and the graph is a line through the origin.

Step 1: Understanding the Problem

To find the intercepts of the equation, we'll look for both the point where the graph of the equation crosses the x-axis (x-intercept) and where it crosses the y-axis (y-intercept). The given equation is a linear equation, \(y = x\), which represents a straight line.

Step 2: Finding the X-Intercept

The x-intercept occurs where \(y = 0\). Substitute \(y = 0\) into the equation \(y = x\) and solve for \(x\). \[ 0 = x \]Thus, the x-intercept is \(x = 0\). So the x-intercept is at the point \((0, 0)\).

Step 3: Finding the Y-Intercept

The y-intercept occurs where \(x = 0\). Substitute \(x = 0\) into the equation \(y = x\) and solve for \(y\). \[ y = 0 \]Thus, the y-intercept is \(y = 0\). So the y-intercept is at the point \((0, 0)\).

Step 4: Graphing the Equation

Now that we have both intercepts, we can plot the graph. Since the equation \(y = x\) is a straight line with the intercepts at \((0, 0)\), the line passes through the origin. For additional points, you can choose any value for \(x\), and the same value is \(y\), such as \((1, 1), (-1, -1)\). Draw a straight line through these points.

Key Concepts

X-Intercept

The x-intercept of a linear equation is the point at which the graph of the equation crosses the x-axis. To find the x-intercept, you need to set the value of y to zero and solve the equation for x. This is because the x-intercept occurs when the output value, y, is zero.

For the equation given in our example, which is a simple linear equation, \(y = x\), finding the x-intercept is straightforward:

• Substitute \(y = 0\) in the equation: \[0 = x\]
• Solve for x: the solution is \(x = 0\).

Thus, the x-intercept for this line is at the origin, which is the point \((0, 0)\). Remember, the x-intercept tells you the specific point on the x-axis that the line touches.

Y-Intercept

The y-intercept of a linear equation indicates where the line crosses the y-axis. To find it, set x equal to zero and solve the equation for y. The y-intercept is significant because it allows you to understand how the line begins on the y-axis.

Let's see how to find the y-intercept with our example equation, \(y = x\):

• Substitute \(x = 0\) into the equation: \[y = 0\]
• Solving gives \(y = 0\).

Hence, the y-intercept is also at the point \((0, 0)\). This means that the line passes through the same point on the y-axis as it does on the x-axis, which is the origin. Every time you're working with a y-intercept, you're finding the point where the line meets the y-axis.

Graphing

Graphing a linear equation, such as \(y = x\), involves plotting points on a graph and drawing a line through them. Understanding intercepts plays a key role in this process since these points help illustrate where the line starts with respect to each axis.

In our example, both intercepts are at the origin, \((0, 0)\), which provides a common starting point. For graphing:

• Plot the intercept point at \((0, 0)\).
• Choose additional points for clarity: for \(x = 1\), \(y = 1\), so plot \((1, 1)\); for \(x = -1\), \(y = -1\), so plot \((-1, -1)\).
• Once these points are plotted, draw a straight line through them.

The resulting graph is a diagonal line passing through the origin, perfectly bisecting the quadrants. Every point on this line satisfies the equation \(y = x\) because it represents equal x and y values. Graphing visually confirms the relationship presented by the equation.