Problem 11 - Understanding Elementary Algebra with Geometry

Question

Find the \(x\) - and \(y\) -intercepts of the equation. $$y-x=-4$$

Step-by-Step Solution

Answer

(4, 0) is the x-intercept, and (0, -4) is the y-intercept.

Step 1: Identify the equation

The given equation is \(y - x = -4\).

Step 2: Find the x-intercept

To find the x-intercept, set \(y = 0\) and solve for \(x\). Substitute \(y = 0\) into the equation: \(0 - x = -4\)Simplify: \(x = 4\)Thus, the x-intercept is at \((4, 0)\).

Step 3: Find the y-intercept

To find the y-intercept, set \(x = 0\) and solve for \(y\). Substitute \(x = 0\) into the equation: \(y - 0 = -4\)Simplify: \(y = -4\)Thus, the y-intercept is at \((0, -4)\).

Key Concepts

x-intercept

The concept of the x-intercept is vital in understanding linear equations. The x-intercept is the point where a line crosses the x-axis on a graph. To find the x-intercept of a linear equation, we set the value of y to 0 and solve for x.

In the given equation \( y - x = -4 \), we set \( y = 0 \).
By substituting \( 0 \) for \( y \), we get:
\[ 0 - x = -4 \]
Solve this simple equation to find \( x \):
\[ x = 4 \]
Thus, the x-intercept is at the point \( (4, 0) \).

Finding the x-intercept is helpful because it shows where the line intersects the x-axis, giving you one fixed point through which the line passes. This point can be useful in plotting the graph of the equation.

y-intercept

Similar to the x-intercept, the y-intercept is the point where a line crosses the y-axis on a graph. To find the y-intercept of a linear equation, we set the value of x to 0 and solve for y.

In the given equation \( y - x = -4 \), we set \( x = 0 \).
By substituting \( 0 \) for \( x \), we get:
\[ y - 0 = -4 \]
Simplify to find \( y \):
\[ y = -4 \]
Thus, the y-intercept is at the point \( (0, -4) \).

Finding the y-intercept is handy for plotting the line on a graph, as it gives a clear vertical point that the equation passes through. Together with the x-intercept, the y-intercept helps in accurately sketching the line represented by the linear equation.

solving linear equations

Solving linear equations is a fundamental skill in algebra. It involves finding the values of variables that make the equation true. To solve for the intercepts in a linear equation, specific steps can be followed.

For the x-intercept:

• Set \( y = 0 \)
• Solve the equation for x

For the y-intercept:

• Set \( x = 0 \)
• Solve the equation for y

In our given example, \( y - x = -4 \), these steps make it clear how to derive the intercepts.

First, we find the x-intercept by setting \( y \) to 0:
\[ 0 - x = -4 \]
Solving for \( x \):
\[ x = 4 \]
Next, the y-intercept by setting \( x \) to 0:
\[ y - 0 = -4 \]
Simplifying to find \( y \):
\[ y = -4 \]
By following these steps, you can solve any linear equation and find its intercepts, making the plotting and understanding of the equation more straightforward.