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Chapter 2: Problem 85

Find the slope and y-intercept of each line. Graph the line. $$ x+2 y=4 $$

Short Answer

Expert verified
The slope is \(-\frac{1}{2}\) and the y-intercept is 2.

Step by step solution

01

Solve for y

Start by solving the equation for y. The given equation is \(x + 2y = 4\). To solve for y, isolate y on one side of the equation: \(2y = 4 - x\)Divide both sides by 2: \( y = 2 - \frac{x}{2} \)
02

Identify the slope and y-intercept

The equation is now in the slope-intercept form \(y = mx + b\), where m is the slope and b is the y-intercept. From the equation \( y = -\frac{1}{2}x + 2 \), the slope (m) is \( -\frac{1}{2} \) and the y-intercept (b) is 2.
03

Graph the line

To graph the line, start by plotting the y-intercept (0, 2) on the graph. Then, use the slope to find another point. The slope of \( -\frac{1}{2} \) means that for each one unit increase in x, y decreases by half a unit. Starting from (0, 2), move one unit to the right (to 1) and half a unit down to reach the point (1, 1.5). Connect these two points with a straight line to see the graph of the equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

linear equations
Linear equations are equations of the first degree. This means that they involve variables raised to the power of one. Typically, they are in the form of \(Ax + By = C\), where A, B, and C are constants. In these equations, the graph forms a straight line when plotted on a coordinate plane. Solving linear equations involves finding the value(s) of the variable(s) that make the equation true. To do this with our example \(x + 2y = 4\), you need to isolate y to bring it into a more useful form for graphing, called the slope-intercept form.
graphing lines
Graphing lines involves plotting points on a coordinate plane to represent the solutions of a linear equation. Each point on the line is a solution to the equation. To graph our example line \(x + 2y = 4\), you convert the equation into slope-intercept form \(y = mx + b\). This format, \(y = -\frac{1}{2}x + 2\), tells you that the y-intercept is 2 and the slope is -1/2. Start by plotting the y-intercept (0, 2). From this point, you use the slope to find more points on the line. Since the slope is -1/2, for every one unit increase in x, y decreases by half a unit. Connect these points to draw the line.
slope-intercept form
The slope-intercept form of a linear equation is \(y = mx + b\). This form is particularly useful because it immediately gives you two crucial pieces of information: the slope (m) and the y-intercept (b). The slope represents the steepness of the line and the direction it tilts. The y-intercept is the point where the line crosses the y-axis. In our example \(y = -\frac{1}{2}x + 2\), the slope is -1/2, which tells us that the line declines as it moves from left to right. The y-intercept is 2, meaning the line crosses the y-axis at (0, 2). By understanding these two components, you can easily graph the line and understand its behavior on the coordinate plane.

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