Problem 13 - Precalculus

Question

The graph of each equation is a straight line. Graph the equation after finding the \(x\)-and the \(y\) -intercepts. (since you are given that the graph is a line, you need only plot two points before drawing the line.) $$x+y=1$$

Step-by-Step Solution

Answer

The line intercepts the x-axis at \((1, 0)\) and the y-axis at \((0, 1)\).

Step 1: Find the x-intercept

To find the x-intercept, we set \( y = 0 \) in the equation \( x + y = 1 \). Substituting, we have \( x + 0 = 1 \), which simplifies to \( x = 1 \). Thus, the x-intercept is the point \( (1, 0) \).

Step 2: Find the y-intercept

To find the y-intercept, we set \( x = 0 \) in the equation \( x + y = 1 \). Substituting, we get \( 0 + y = 1 \), which simplifies to \( y = 1 \). Therefore, the y-intercept is the point \( (0, 1) \).

Step 3: Plot the intercepts on the graph

On graph paper or a coordinate plane, plot the points \( (1, 0) \) and \( (0, 1) \). These are the x-intercept and y-intercept, respectively.

Step 4: Draw the line

Using a ruler, draw a straight line that passes through the plotted points \( (1, 0) \) and \( (0, 1) \). This line represents the graph of the equation \( x + y = 1 \).

Key Concepts

x-intercept

The x-intercept of a line is the point where the line crosses the x-axis. At this point, the value of y is always zero because it's located directly on the horizontal axis. To find the x-intercept from an equation like \( x + y = 1 \), you simply set \( y = 0 \) and solve for \( x \).

Here's how to do it step-by-step:

• Set \( y = 0 \) in the equation: \( x + 0 = 1 \).
• This simplifies to \( x = 1 \).

So, the x-intercept is the point \((1, 0)\).

This intercept is crucial in graphing because it provides a precise point on the graph, helping you draw the line accurately.

y-intercept

Similarly, the y-intercept is where a line meets the y-axis. Here, the x value is zero because the point lies on the vertical axis. To find the y-intercept of \( x + y = 1 \), set \( x = 0 \) in the equation.

Follow these steps:

• Replace \( x \) with 0 in the equation: \( 0 + y = 1 \).
• Solving this gives \( y = 1 \).

Thus, the y-intercept is \((0, 1)\).

Finding this intercept is essential because, along with the x-intercept, it provides enough information to graph the line without further complex calculations.

graphing lines

Graphing lines using the intercepts is both efficient and straightforward. Once you have both the x-intercept and the y-intercept, plotting and drawing the line is just a few steps away.

Here's the process:

• Plot the x-intercept \((1, 0)\) on the coordinate plane.
• Then, plot the y-intercept \((0, 1)\).
• Use a ruler to draw a line through the two intercept points.

This line represents the equation \( x + y = 1 \).

By graphing with intercepts, you visualize how the equation behaves as a line. This approach is quick and allows you to clearly see where the line crosses both axes, building an intuitive understanding of linear equations.