Problem 49 - Algebra 1

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 equation \(y + 2x = 2\) is -2, and the y-intercept is 2. The graph of the line starts at point (0,2) on the y-axis. From there, the line moves down 2 units and right 1 unit for each step due to the slope of -2/1.

Step 1: Arrange Equation to Slope-Intercept Form

First, arrange the given equation \(y + 2x = 2\) in the form of \(y = mx + b\). For this, subtract \(2x\) from both sides to isolate \(y\). This would result in \(y = -2x + 2\).

Step 2: Identify the Slope and Y-intercept

With the equation \(y = -2x + 2\), it is clear that \(m=-2\), which is the slope of the equation, and \(b=2\), which is the y-intercept.

Step 3: Plotting the Equation

To plot the graph for the equation, firstly mark the y-intercept point at \(y = 2\) on the y-axis. Then, use the slope to determine the next point. Since the slope is -2, which can also be written as -2/1, start at the y-intercept and move 2 units downward (negative sign indicates downward movement) and 1 unit in the positive x direction (denominator indicates rightward movement). Repeat this process several times to establish a series of points, then connect the points to form a straight line.

Key Concepts

Understanding Slope

The slope of a line is a measure of its steepness and direction. It's commonly represented by the letter \(m\) in the slope-intercept form of a linear equation \(y = mx + b\). The slope tells us how much the \(y\) coordinate of a point on the line will change as the \(x\) coordinate increases by one unit. A positive slope means the line rises as it moves from left to right, while a negative slope indicates the line falls.

For example, a slope of \(-2\) means that for every one unit you move to the right on the x-axis, the line moves two units down on the y-axis.

• A slope of 0 results in a horizontal line where the \(y\) coordinate remains constant.
• An undefined slope corresponds to a vertical line, where the \(x\) coordinate is constant.

Understanding the slope is crucial for predicting how changes in the x values affect the y values on a graph. This understanding is central to topics like linear motion, economics, and biology, where rate of change is a key concept.

The Role of the Y-Intercept

The y-intercept is where a line crosses the y-axis. It is symbolized by \(b\) in the slope-intercept form \(y = mx + b\). This value tells us the point at which the line intersects the y-axis when \(x=0\).

In the equation \(y = -2x + 2\), the y-intercept is \(2\). This means that when \(x\) is zero, \(y\) will be \(2\). This point is often the starting point for graphing a line because the y-intercept is a fixed point.

• This is helpful in real-world scenarios such as finding initial costs in a budget graph, where every unit of production might start with a base cost even when no items are produced.
• It gives a clear visual on the graph to signal the start point of a line, aiding in drawing and interpreting linear models.

Understanding the y-intercept helps provide a contextual starting point, making it easier to graph linear equations and to understand information at zero units of a variable.

Graphing Linear Equations

Graphing linear equations is the process of drawing a line that represents all possible solutions of an equation on a coordinate plane. One of the simplest forms to graph from is the slope-intercept form, \(y = mx + b\), as it directly reveals the slope and y-intercept.

To graph the equation \(y = -2x + 2\):

• Start at the y-intercept (0, 2) and make a mark on the y-axis.
• Use the slope, \(-2\), to find the next points: from the y-intercept, move down 2 units and to the right 1 unit to get to the next point.
• Continue plotting additional points using the slope until you have a few on the graph.
• Draw a straight line through these points, extending it across the graph.

This linear graph gives a visual representation of the relationship between \(x\) and \(y\) in the equation. By using the slope and the y-intercept, you gain a quick and efficient way to sketch a graph, which is useful for both academic exercises and real-world problem-solving where time and precision are of the essence.