Problem 68 - Algebra 1: Concepts and Skills

Question

Find the slope and the y-intercept of the graph of the equation. Then graph the equation. $$ y+2 x=2 $$

Step-by-Step Solution

Answer

The slope of the line is -2 and the y-intercept is at the point (0,2).

Step 1: Transformation to Slope-Intercept Form

First, we need to rearrange the given equation \(y+2x=2\) into the slope-intercept form. This can be done by subtracting \(2x\) from both sides of the equation to isolate \(y\). Therefore, the equation becomes \(y=2-2x.\)

Step 2: Identify the Slope and the Y-Intercept

From the equation \(y = 2 - 2x\), we can identify the slope \(m\) and the y-intercept \(b\). Here, the slope \(m\) is -2 (the coefficient of \(x\)) and the y-intercept \(b\) is 2 (the constant term).

Step 3: Graphing the Equation

To graph the equation, we start by plotting the y-intercept at (0,2). Then, we use the slope to find another point on the line: because the slope is -2, this means that for each 1 unit increase in \(x\), the value of \(y\) decreases by 2 units. Therefore, from the point (0,2), moving 1 unit to the right for \(x\) results in a decrease in \(y\) by 2 units, leading to the point (1,0). We then plot these points and draw a straight line passing through them.

Key Concepts

Slope-Intercept Form

The **slope-intercept form** is a special way of writing the equation of a straight line. It is written as \( y = mx + b \), where \(m\) represents the slope of the line and \(b\) is the y-intercept. This form makes it very straightforward to graph a line and understand how it behaves.
To transform any linear equation into slope-intercept form, you need to solve for \(y\) and isolate it on one side of the equation. This helps in clearly identifying both the slope and the y-intercept for graphing purposes. For example, if the original equation is \( y + 2x = 2 \), rearranging it by subtracting \(2x\) gives you \( y = 2 - 2x \), which is now in slope-intercept form.

• Slope \(m\): The coefficient of \(x\).
• Y-intercept \(b\): The constant term.

Using slope-intercept form makes it easier to draw the line on a graph because it clearly shows how much \(y\) changes with \(x\).

Slope of a Line

The **slope of a line** is an important concept that indicates the steepness and direction of a line. It is denoted by \(m\) and can be found from the slope-intercept form \( y = mx + b \) as the coefficient of \(x\).
Slope is calculated as the "rise over run," meaning how much \(y\) increases or decreases when \(x\) changes by 1 unit. In other words, it shows the vertical change per unit of horizontal change.
For example, in the line equation \( y = 2 - 2x \), the slope is \(-2\). This slope tells us that for every 1 unit \(x\) increases, \(y\) decreases by 2 units. The negative slope also indicates that the line goes downward from left to right. Positive slopes go upwards, and zero slope represents a flat, horizontal line.

• If \(m > 0\), the line rises.
• If \(m < 0\), the line falls.
• If \(m = 0\), the line is horizontal.

Y-Intercept

The **y-intercept** of a line is where the line meets the y-axis. It is represented as \(b\) in the equation \( y = mx + b\). This point has a special characteristic: the \(x\)-value is always zero because it is the place where the line crosses the y-axis, effectively acting as an anchor for graphing the line.
In the equation \( y = 2 - 2x \), the y-intercept is 2. This means if you start graphing on the coordinate plane, your line will cross the y-axis at the point \( (0, 2) \).

• The y-intercept gives you a starting point for your graph.
• Once you have this point, you can use the slope to find other points on the line.

Being able to accurately identify the y-intercept is key for drawing the line correctly and understanding its features.

Graphing Linear Equations

**Graphing linear equations** involves using the slope and y-intercept to draw a line on a coordinate plane. Equipped with the equation in slope-intercept form, such as \( y = -2x + 2 \), you can easily proceed to visualize the line.
Start by marking the y-intercept on the graph. In this example, plot the point \( (0, 2) \) since the y-intercept is 2. Then, apply the slope to determine the direction and steepness of the line. With a slope of -2, you should descend 2 units for every 1 unit you move to the right along the x-axis.

• Begin at the y-intercept point.
• Move right along the x-axis. Use the slope to determine the next point.
• If the slope is negative, go downwards; if it's positive, go upwards.

Once you have at least two points, draw a straight line through them to extend across the grid. This line represents your linear equation graphically, clearly showing the relationship between \(x\) and \(y\). Understanding this process helps in accurately predicting and visualizing different situations that can be represented through linear equations.