Question

In Exercises \(27-30,\) find the (a) slope and (b) \(y\) -intercept, and (c) graph the line. $$x+y=2$$

Short answer

The slope of the line is -1, the y-intercept is 2, and the graph is a straight line passing through the y-intercept (0,2) and sloping downwards from left to right.

Step-by-step solution

Find Slope

Rewrite the given equation to the form \(y = mx + c\). This gives \(y = -x + 2\). The coefficient of \(x\) is the slope, so the slope \(m\) is \(-1\).

Find y-intercept

The y-intercept is the constant term in the equation, which is \(2\)

Graph the Line

Start by plotting the y-intercept (0, 2) on the graph. Since the slope is \(-1\), it means that for every step to the right on the x-axis, we go one step down. Conversely, for each step left, we go one step up. Draw the line that passes through these points.

Key concepts

Slope of a Line

Understanding the slope of a line is fundamental in graphing linear equations. The slope is a measure of how steep a line is. Mathematically, it's defined as the rate of change of the y-coordinate as the x-coordinate increases. In simpler terms, it tells you how much the y value rises or falls as x moves left to right.

When looking at the equation of a line in the form \( y = mx + c \), the slope is represented by the coefficient \( m \). If the slope is positive, the line rises as it moves from left to right. Conversely, if the slope is negative, the line falls. For the given exercise, the slope, \( m \) of the line \( x+y=2 \) is \( -1 \), indicating a line that descends to the right at a 45-degree angle, as each step horizontally corresponds to an equivalent step downward.

Y-Intercept

The y-intercept is another crucial concept in the graphing of linear equations. It represents the point where the line crosses the y-axis. This can be found by evaluating the linear equation when \( x \) is zero. In our standard form equation \( y = mx + c \), the y-intercept is denoted by \( c \).

For a given line, identifying the y-intercept allows for an easy starting point to begin graphing. It's the value of y when x equals zero. In the example equation \( x + y = 2 \), rearranging to \( y = -x + 2 \) reveals the y-intercept to be \( 2 \). This signifies that the line will cross the y-axis at the point (0, 2).

Linear Equation Graphing

Graphing a linear equation involves plotting points on the coordinate plane and connecting them to form a straight line. Begin by locating the y-intercept on the y-axis which, as mentioned earlier, is the point where the line crosses the y-axis. Next, use the slope to find another point. Each unit of the slope indicates a move in the horizontal (\(x\) direction) and vertical (\(y\) direction) on the graph.

In our exercise, we start at the y-intercept (0,2), and because the slope is -1, for every unit we move right, we move one unit down. Plotting another point, say at \( x = 1 \), would give us the point (1,1) by decrementing the y value, in line with the slope's instruction. Connecting these points with a straight edge gives us the graph of the equation.

Standard Form of a Linear Equation

The standard form of a linear equation is expressed as \( Ax + By = C \), where \( A \), \( B \), and \( C \) are integers, and \( A \) and \( B \) are not both zero. This form is useful for quickly determining several properties of the line, such as its x and y-intercepts. To graph an equation in standard form, you may need to rearrange it into slope-intercept form \( y=mx+c \) for more convenience.

In the example problem, the equation \( x + y = 2 \) is already in standard form. By rearranging it to \( y = -x + 2 \), we converted it to the slope-intercept form, making it easier to identify the slope and y-intercept for graphing. Understanding how to manipulate linear equations between these forms is an important skill in algebra.