Problem 74 - Algebra 1

Question

Write the equation in slope-intercept form. Then graph the equation. $$ x+2 y-6=0 $$

Step-by-Step Solution

Answer

The equation in slope-intercept form is \(y = -0.5x + 3\). The graph of this equation starts at the point (0,3) and slopes downward at a gradient of -0.5.

Step 1: Convert to Slope-Intercept Form

The equation can be converted into the slope-intercept form by rearranging the equation: \(x+2y-6=0\) to \(2y = -x + 6\), and then divide by 2 to find \(y\), giving \(y = -0.5x + 3\).

Step 2: Identify the Slope and Y-intercept

From the equation \(y = -0.5x + 3\), it can be observed that the slope 'm' is -0.5 and the y-intercept 'b' is 3.

Step 3: Graph the Equation

Begin by plotting the y-intercept (0,3). From here, take the slope (-0.5) which means when 'x' increases by 1, 'y' decreases by 0.5. Continue this process until you have enough points to make a graph. Sketch the line that runs through these points.

Key Concepts

Graphing Linear Equations

Graphing linear equations is a fundamental concept in algebra. Here, the goal is to visualize the relationship between variables on a coordinate plane.
Linear equations are often expressed in slope-intercept form, which is \(y = mx + b\). This form readily tells you the slope and the y-intercept, two key pieces of information for graphing.

To graph a linear equation, follow these steps:

• First, write the equation in slope-intercept form if it isn't already. For instance, rearrange terms and solve for 'y'.
• Identify the y-intercept: where the line crosses the y-axis, marked by 'b'.
• Determine the slope 'm'. This will dictate how to move from the y-intercept to plot additional points on the graph.
• Use the slope to plot more points: move right 1 unit, and then up or down according to the slope.
• Draw a straight line through all points, extending across the graph.

Understanding this visual process helps in better interpreting and analyzing the relationships represented by linear equations.

Slope

The slope of a line is a measure of its steepness and direction. In mathematical terms, it is the ratio of the vertical change to the horizontal change between two points on a line.

The slope is denoted by 'm' in the slope-intercept equation \(y = mx + b\). A few important facts about slope:

• Positive slope: The line inclines upwards as it moves from left to right.
• Negative slope: The line declines downwards as it moves from left to right.
• Zero slope: Represents a horizontal line.
• Undefined slope: Represents a vertical line.

In our example \(y = -0.5x + 3\), the slope is -0.5. This means for every 1 unit increase in x, y decreases by 0.5 units. Understanding slope is critical in determining how lines are oriented and predicting their behavior on a graph.

Y-intercept

The y-intercept is where the line crosses the y-axis on a graph. It's a crucial point because it provides a starting point for graphing the rest of the line.
The y-intercept is denoted by 'b' in the slope-intercept form \(y = mx + b\).

• This point is always represented as (0, b), where 'b' is the y-intercept value.
• It's the value of 'y' when 'x' is zero.

For example, in the equation \(y = -0.5x + 3\), the y-intercept is 3. This point (0, 3) is plotted first on the y-axis and forms the starting point for graphing the line. Identifying the y-intercept immediately provides a foundation to graph linear equations accurately and helps to understand the initial condition of the line.