Problem 46 - Algebra 1

Question

Find the \(x\) -intercept and the \(y\) -intercept of the line. Graph the equation. Label the points where the line crosses the axes. $$ y=4 x+8 $$

Step-by-Step Solution

Answer

The x-intercept is -2 and the y-intercept is 8. The line intersects the axes at these points.

Step 1: Find the x-intercept

Set \(y = 0\) in the given equation and solve for \(x\):\n\n0 = 4x + 8\n\nThis simplifies to \(x = -2\). So, the x-intercept is -2.

Step 2: Find the y-intercept

Set \(x = 0\) in the given equation and solve for \(y\):\n\ny = 4(0) + 8\n\nThis simplifies to \(y = 8\). So, the y-intercept is 8.

Step 3: Graph the equation

Now graph the line using the x and y intercepts. The line will cross the x-axis at \(x = -2\) and the y-axis at \(y = 8\). Label these points on the graph.

Key Concepts

Understanding the x-intercept

The x-intercept of a line is the point where the line crosses the x-axis. At this location, the y-value is always zero. To find the x-intercept in a linear equation like the given one (\(y = 4x + 8\)), we set \(y = 0\) and solve for \(x\).

Here's how it works step-by-step:

• Start with the equation: \(0 = 4x + 8\).
• Subtract 8 from both sides to isolate the term with \(x\): \(0 - 8 = 4x\). This simplifies to \(-8 = 4x\).
• Divide both sides by 4 to solve for \(x\): \(x = -2\).

So, the x-intercept is \((-2, 0)\). This tells us that the line crosses the x-axis at \(x = -2\). Knowing how to find the x-intercept is crucial because it gives us one of the points we need to draw the line on a graph. This method applies consistently to linear equations, making it a reliable tool in your math arsenal.

Exploring the y-intercept

The y-intercept is the point where a graph crosses the y-axis. This occurs when \(x\) is zero. For any linear equation, finding the y-intercept involves substituting \(x = 0\) into the equation and then solving for \(y\). In the equation \(y = 4x + 8\), this process is simple.

Follow these steps:

• Plug \(x = 0\) into the equation: \(y = 4(0) + 8\).
• This simplifies to \(y = 8\).

The y-intercept is \((0, 8)\), meaning our line hits the y-axis at \(y = 8\). This point is essential because it provides a starting position on the graph when you're plotting the line. Unlike the x-intercept, which is found by setting \(y\) to zero, for the y-intercept we always look at where \(x\) equals zero.

Graphing lines: Putting it all together

Graphing lines using intercepts is an effective method because it only requires two key points. Once you have these points, you can draw a straight line through them. Consider our findings for the line \(y = 4x + 8\): we have the x-intercept at \((-2, 0)\) and the y-intercept at \((0, 8)\).

To graph the line:

• Start by plotting the x-intercept. Mark the point \((-2, 0)\) on the x-axis.
• Next, plot the y-intercept. Mark the point \((0, 8)\) on the y-axis.
• Draw a straight line connecting these two points extending it on both ends. This line represents all points \(\{(x, y)\}\) that satisfy the equation \(y = 4x + 8\).

Graphing lines like this visually confirms the relationship between \(x\) and \(y\) in the equation. It's a practical way to understand linear equations, see their slope, and even predict values. Always remember to label your intercepts on the graph clearly, as this will help you and others recognize key features of the line at a glance.