Problem 94 - Algebra 1

Question

Find the \(x\) -intercept and the \(y\) -intercept of the graph of the equation. $$y=-2 x+16$$

Step-by-Step Solution

Answer

The x-intercept of the graph is \(x=8\) and the y-intercept of the graph is \(y=16\).

Step 1: Find the x-intercept

Set \(y = 0\) in the equation and solve for \(x\). This gives the equation \(0=-2x+16\). If you solve it, this leads to \(x=8\). Therefore, the x-intercept is \(x = 8\).

Step 2: Find the y-intercept

Set \(x = 0\) in the equation and solve for \(y\). This gives the equation \(y=-2*0+16\). The answer is \(y = 16\). Therefore, the y-intercept is \(y = 16\).

Key Concepts

Understanding the X-Intercept

The concept of the x-intercept is quite fundamental when working with linear equations. To determine where a line crosses the x-axis, we set the value of \( y \) to zero.
Let's look at the equation from the exercise: \( y = -2x + 16 \).
By placing \( 0 \) for \( y \), you get the equation \( 0 = -2x + 16 \). To isolate \( x \), you solve the equation:

• First, move \( 16 \) to the other side by subtracting 16 from both sides, resulting in \( -16 = -2x \).
• Then, divide both sides by \(-2\) to solve for \( x \). This gives \( x = 8 \).

The x-intercept is the point \((8, 0)\), indicative of where the line graph meets the x-axis on a coordinate plane.

Exploring the Y-Intercept

Finding the y-intercept involves a similar process but in reverse. This is where the line intersects the y-axis. To find it, set \( x = 0 \) and solve for \( y \).
Using the exercise's equation, \( y = -2x + 16 \), substitute \( 0 \) for \( x \), leaving \( y = -2(0) + 16 \).
Because multiplying by zero results in zero, the calculation simplifies to \( y = 16 \).
Thus, the y-intercept is at the point \((0, 16)\).

• This point shows precisely where the graph intersects the y-axis.
• Having both intercepts is crucial for graphing a line on a coordinate plane.

Graphing Linear Equations

Graphing a linear equation like \( y = -2x + 16 \) requires understanding of both intercepts. They are crucial points of reference. With the x-intercept \((8, 0)\) and the y-intercept \((0, 16)\), you can plot these points on a coordinate plane to visually represent the equation.
After plotting, simply draw a straight line through these points. This line embodies all solutions of the equation.
Key points to remember:

• Linear equations form straight lines on graphs. Knowing the intercepts provides straightforward points to draw from.
• A graph can give a quick insight into how variables relate visually, offering a clearer understanding of the equation's behavior.
• Intercepts are not just numbers; they are coordinates that guide your graphing process.

Finding and using intercepts are fundamental tasks in graphing linear equations taught in algebra.