Problem 16 - Elementary Algebra

Question

For Problems 1-36, graph each linear equation. (Objective 2) $$ y=-2 x $$

Step-by-Step Solution

Answer

The graph is a line through the origin with a slope of -2, passing through points like (0,0) and (1,-2).

Step 1: Identify the Equation Type

The given equation is a linear equation in the form of \(y = mx + c\). Here, \(m = -2\) and \(c = 0\). This means the equation is in the slope-intercept form \(y = mx + b\), where \(b\) is the y-intercept.

Step 2: Determine the Slope and Y-intercept

The slope of the line is \(-2\) and the y-intercept is \(0\). This means the line will pass through the origin \((0, 0)\) and for every one unit increase in \(x\), \(y\) will decrease by \(2\).

Step 3: Plot the Y-Intercept

Start by plotting the y-intercept point \((0, 0)\) on the graph. This is where the line will cross the y-axis.

Step 4: Use the Slope to Plot Another Point

From the y-intercept \((0, 0)\), use the slope \(-2\). This means you will go down 2 units in the y direction as you move 1 unit to the right in the x direction. This leads to another point \((1, -2)\) which you can plot on the graph.

Step 5: Draw the Line

Draw a straight line through the points \((0, 0)\) and \((1, -2)\). Extend this line across the graph to show the full extent of the equation. Make sure the line is straight and goes through all points determined by the slope \(-2\).

Key Concepts

Graphing Linear Equations

Graphing linear equations involves visually representing them on a coordinate plane. It's a method that shows the solutions of an equation as a line. For the equation \(y = -2x\), we're observing how changes in \(x\) affect \(y\). The line on a graph represents all combinations of \(x\) and \(y\) that satisfy the equation.
To understand graphing, follow these steps:

• Identify the equation's form. Here, it's a straight line since it is linear.
• Determine key features like the slope and y-intercept, which guide you in plotting the graph.
• Plot points using the slope to see where the line will go on the plane.

Graphing helps in visualizing relationships between variables, making equations easier to interpret.

Slope-Intercept Form

The slope-intercept form is a straightforward way to write linear equations. It's given by \(y = mx + b\), where \(m\) represents the slope and \(b\) is the y-intercept. This form is powerful in simplifying the graphing process.
In our equation \(y = -2x\):

• Here, \(m = -2\). This tells us how steep the line is and in which direction it moves as \(x\) changes.
• \(b = 0\), meaning the line crosses the y-axis at the origin.

Understanding this form helps quickly identify characteristics of the line, such as its starting point and angle of tilt.

Plotting Points

Plotting points is an essential step in drawing a line on a graph. It involves determining exact locations on the coordinate plane that the line passes through.
To plot points for the equation \(y = -2x\):

• Start with the y-intercept \((0, 0)\) since this is where the line will intersect the y-axis.
• Using the slope, move 1 unit right (increasing \(x\)) and 2 units down (decreasing \(y\)) to find the next point, \((1, -2)\).
• Repeat this step to plot additional points if needed for accuracy.

Connected points illustrate the linear path that the equation describes. Always check for alignment to ensure accuracy in graphing.

Y-Intercept

The y-intercept is a fundamental concept in understanding where a line crosses the y-axis on a graph. It is the value of \(y\) when \(x = 0\).
For the line \(y = -2x\):

• The y-intercept is \((0, 0)\). This shows that the graph line begins at the origin.

The y-intercept is crucial because it serves as a starting point for plotting the graph. Knowing this allows quicker and more accurate placement of the line on a coordinate plane, as you already have one definitive point to begin plotting from.