Problem 29 - Elementary Algebra

Question

For Problems 1-36, graph each linear equation. (Objective 2) $$ -2 x+y=-4 $$

Step-by-Step Solution

Answer

Graph the line with y-intercept -4 and slope 2.

Step 1: Identify the Slope-Intercept Form

We need to graph the equation, and a good starting point is to write it in slope-intercept form, which is \( y = mx + b \). The given equation is \( -2x + y = -4 \).

Step 2: Solve for y

Rearrange the equation to solve for \( y \). Add \( 2x \) to both sides to isolate \( y \): \( y = 2x - 4 \). Now the equation is in slope-intercept form.

Step 3: Identify the Slope and Y-Intercept

In the equation \( y = 2x - 4 \), the slope \( m \) is 2 and the y-intercept \( b \) is -4. This means the line will rise 2 units vertically for every 1 unit it moves horizontally, and it crosses the y-axis at -4.

Step 4: Plot the Y-Intercept

Plot the point (0, -4) on the graph. This is where the line crosses the y-axis.

Step 5: Use the Slope to Find Another Point

Starting from the y-intercept (0, -4), use the slope to find another point. From (0, -4), move up 2 units and right 1 unit to reach the point (1, -2). Plot this point.

Step 6: Draw the Line

Draw a line through the two points (0, -4) and (1, -2). Extend this line across the graph to represent the linear equation.

Key Concepts

Slope-Intercept Form

When graphing a linear equation, the first step is often to get the equation into slope-intercept form. This format is \( y = mx + b\).
The reason it's so useful is because it transparently shows two crucial features of the line: the slope \( m \) and the y-intercept \( b \). The equation becomes easy to interpret once it's in this form.

• \( m \) - the slope, which tells you how steep the line is.
• \( b \) - the y-intercept, where the line crosses the y-axis.

For the equation \(-2x + y = -4\), converting to slope-intercept form involves rearranging to solve for \( y \). This gives us \( y = 2x - 4 \). Now, the line’s slope and where it intersects the y-axis are clear.

Plotting Points

Plotting points is an integral part of graphing linear equations, as it helps in pinpointing exact locations on a graph to draw the line.
To correctly graph a line, you must start with known points, usually beginning with the y-intercept.

• Start by plotting the y-intercept (0, -4) based on the slope-intercept equation \( y = 2x - 4 \).
• Use the slope to identify a second point: from (0, -4), move up 2 units and to the right by 1 unit, arriving at the point (1, -2).

After placing these points on the graph, connect them with a straight line, ensuring the line continues across the graph’s bounds. This method ensures accuracy when representing the linear equation visually.

Linear Equations

Linear equations represent straight lines on a graph and are foundational in algebra. They describe a constant rate of change, which is depicted by the slope.
For example, our equation \( y = 2x - 4 \) represents a line where any change in \( x \) by 1 unit leads to a change in \( y \) by 2 units. Linear equations are straight-forward because they only require a basic set of operations: addition, subtraction, multiplication, and division.
These operations maintain the equation's balance while allowing you to identify both the slope and intercept.

Slope and Y-Intercept

The slope and y-intercept are key components of any linear equation, determining the line's direction and starting point.Slope:

• The slope is the term \( m \) in the equation \( y = mx + b \).
• It dictates the line's steepness and direction: positive slopes rise to the right, while negative ones fall.
• Our slope \( m \) is 2, indicating the line climbs 2 units up for every additional unit in the x-axis.

Y-Intercept:

• The y-intercept \( b \) is where the line crosses the y-axis, at point (0, \( b \)).
• In this case, \( b \) is -4, so the line crosses the y-axis at -4.

These two elements make it simple to accurately draw and understand the graph of a linear equation, giving clarity on its trajectory and starting point.