Problem 90 - Algebra 1: Concepts and Skills

Question

Write the equation in slope-intercept form. Then graph the equation. (Lesson 4.7) $$-x+y-7=0$$

Step-by-Step Solution

Answer

The equation in slope-intercept form is \(y = x + 7\). The graph starts at the y-intercept (0,7) and then increases 1 unit up and 1 unit to the right from this point.

Step 1: Rewrite the equation in slope-intercept form

Rearranging the given equation \(-x + y - 7 = 0\) to the form \(y = mx + b\), add x to both sides to isolate the terms involving y on one side of the equation: \[y = x + 7\]. Here, the slope, m, is 1 and the y-intercept, b, is 7.

Step 2: Identify the slope and y-intercept

From the slope-intercept form found on Step 1, the slope (m) is equivalent 1 and the y-intercept (b) equals 7. This means that the line intersects the y-axis at the point (0,7) and slopes upwards as we go from left to right since the slope is positive.

Step 3: Plot the y-intercept

Start the graph by plotting the y-intercept at the point (0,7). This is where the line crosses the y axis.

Step 4: Use the slope to plot the next point

From the y-intercept, use the slope to find the next point. The slope, commonly represented as rise/run or change in y/change in x, tells us to move one unit up (because of the positive slope) and one unit to the right. From the y-intercept (0,7), we would therefore go to the point (1,8).

Step 5: Draw the line

Once two points are plotted on the graph, draw a line through them. This is the graphical representation of the equation in the given exercise.

Key Concepts

Graphing Linear Equations

Understanding how to graph linear equations is a foundational skill in algebra. It involves taking an equation representing a straight line and turning it into a visual graph. The most common form used for graphing is the slope-intercept form, which is expressed as \(y = mx + b\). Here, \(m\) represents the slope of the line, and \(b\) is the y-intercept, the point where the line crosses the y-axis.

When graphing, the first step is to plot the y-intercept on the graph. For the exercise with the equation \( -x + y - 7 = 0 \), it was re-arranged to \( y = x + 7\) to find the y-intercept (\(0, 7\)). Next, the slope, which is the ratio of the vertical change (\

Equation of a Line

The equation of a line provides a way to describe the relationship between two variables, typically \(x\) and \(y\). The most straightforward type of line equation is in slope-intercept form, \(y = mx + b\), which efficiently describes the direction and position of a line on the coordinate plane. Within this equation, \(m\) denotes the slope, showing how steep the line is, and \(b\) indicates the y-intercept, marking where the line crosses the y-axis.

To transform an equation to slope-intercept form, one must isolate \(y\) on one side. Taking the original exercise equation \( -x + y - 7 = 0 \), by rearranging it to the standard slope-intercept form, we get \(y = x + 7\), where the slope is 1 (indicating a 45-degree angle) and the y-intercept is \(7\), telling us the line touches the y-axis at \(7\). This conversion simplifies graphing and analyzing the line's properties.

Slope and Y-intercept

The slope and y-intercept are vital components in the equation of a line and play a key role in understanding its behavior. The slope, \(m\), describes the angle of inclination or the steepness of the line. It is a measure of how much \(y\) changes for a given change in \(x\). Positive values of \(m\) indicate an upward trend as one moves from left to right, while negative values denote a downward trend.

The y-intercept is the second critical component, symbolized by \(b\) in the slope-intercept equation, \(y = mx + b\). It is the specific point where the line crosses the y-axis, providing a reference point for drawing the line on a graph. In the given exercise, the equation \(y = x + 7\) has a slope (\(m\)) of 1, and a y-intercept (\(b\)) of 7. This tells us, starting from the point (0,7) on the y-axis, the line will rise one unit up for every unit it moves to the right.