Problem 6 - Elementary Algebra

Question

For Problems 1-36, graph each linear equation. (Objective 2) $$ -x+y=5 $$

Step-by-Step Solution

Answer

Graph the line with slope 1 and y-intercept 5, passing through points (0,5) and (1,6).

Step 1: Write the Equation in Slope-Intercept Form

The given equation is \(-x + y = 5\). To make it easier to graph, rewrite it in the slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Add \(x\) to both sides to get: \(y = x + 5\). This shows that the slope \(m\) is 1 and the y-intercept \(b\) is 5.

Step 2: Identify the Y-Intercept

The y-intercept is the point where the line crosses the y-axis. From the equation \(y = x + 5\), the y-intercept \(b\) is 5. Thus, the point \((0, 5)\) is on the graph.

Step 3: Use the Slope to Find Another Point

The slope of the equation is 1, which means for each unit you move to the right along the x-axis, you move 1 unit up on the y-axis. Starting at the point \((0, 5)\), move 1 unit right to \(x = 1\) and 1 unit up to \(y = 6\). Plot the point \((1, 6)\) on the graph.

Step 4: Draw the Line

Draw a straight line through the points \((0, 5)\) and \((1, 6)\). Extend this line across the grid, maintaining the slope of 1.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way of writing linear equations so that they are easy to analyze and graph. This form is written as \(y = mx + b\). Here, \(m\) represents the slope of the line, and \(b\) represents the y-intercept. This format allows us to quickly determine the important characteristics of the line and is useful for sketching graphs.

Understanding the components:

• Slope \( (m) \): The slope is a measure of how steep the line is. It tells us the rise over run, meaning how much \(y\) changes for a change in \(x\). A positive slope means the line goes upward from left to right, while a negative slope means it goes downward.
• Y-intercept \( (b) \): This is the point at which the line crosses the y-axis. It tells us the value of \(y\) when \(x\) is 0.

In our exercise, we start with the equation \(-x + y = 5\). By rearranging it to \(y = x + 5\), we see that \(m = 1\) and \(b = 5\). This makes it straightforward to graph the line.

Graphing

Graphing a linear equation means plotting points that satisfy the equation and drawing a line through these points. In our example, we use the slope-intercept form of the equation, \(y = x + 5\), to find points that lie on the line.

Steps for graphing:

• Identify the y-intercept \((0, b)\): Start by plotting the y-intercept on the graph. This is always an easy point to find because \(b\) gives us \(y\) when \(x = 0\).
• Use the slope \((m)\): The slope tells us how to find additional points. For example, if \(m = 1\), it means for each step right along the x-axis, you take one step up on the y-axis.
• Plot another point: Starting at the y-intercept, use the slope to find another point. In our case, starting from \((0, 5)\), moving 1 unit right and 1 unit up takes you to \((1, 6)\).
• Draw the line: Once you have two points, draw a straight line through them. This line is the graph of the equation.

Y-Intercept

The y-intercept is the point where the line crosses the y-axis in the coordinate plane. It's a significant feature because it provides a starting point for the graphing process.

Identifying the y-intercept:

• The y-intercept is given by \((0, b)\). This notation means that when \(x\) is 0, the y-value is \(b\). This point is important because it is easy to locate on the graph.
• From the equation \(y = x + 5\), the y-intercept \(b\) is 5. This means the graph will cross the y-axis at the point \((0, 5)\).
• This point serves as the anchor for drawing the rest of the line, making graphing more straightforward.

Starting from the y-intercept, using the slope, you can graph the entire line efficiently. Remember, the larger the value of \(b\), the higher the line crosses the y-axis.