Problem 16 - Algebra 1: Concepts and Skills

Question

Rewrite the equation in slope-intercept form. $$3 x-6 y=18$$

Step-by-Step Solution

Answer

The equation \(3x - 6y = 18\) written in slope-intercept form is \(y = 0.5x - 3\). The slope of the line is 0.5 and the y-intercept is -3.

Step 1: Identify the Form

Identify the standard form of equation which is \(Ax + By = C\). In the case of the given equation \(3x - 6y = 18\) it can be observed that it has the standard form, where \(A = 3\), \(B = -6\) and \(C=18\).

Step 2: Rewrite in Slope-Intercept Form

Rewrite the equation in slope-intercept form, subtracting 3x from both sides to get \( -6y = -3x + 18\). Then, divide each term by -6 to isolate y and get the equation in the form \(y=mx +c\), which results in \(y = 0.5x - 3\).

Step 3: Identify Slope and Intercept

Now that the equation is in slope-intercept form, the slope (m) and the y-intercept (c) can be determined. The slope (m) is the coefficient of x, which is 0.5, and the y-intercept (c) is -3. The y-intercept is the point where the line crosses the y-axis.

Key Concepts

Standard form of a linear equation

The standard form of a linear equation is a way of expressing linear equations where the equation is structured as \(Ax + By = C\). This format is particularly useful for certain types of mathematical analysis and applications, like finding the intersection points of lines. In the standard form, \(A\), \(B\), and \(C\) are integers, and among these, \(A\) should ideally be a positive number.

• \(A\) and \(B\) are coefficients of the variables \(x\) and \(y\).
• \(C\) represents the constant term.

In the given problem, the equation \(3x - 6y = 18\) fits perfectly into the standard form, where \(A = 3\), \(B = -6\), and \(C = 18\). This form is beneficial to quickly recognize and handle equations before transforming them into other forms like the slope-intercept form.

Slope of a line

The slope of a line is a crucial concept in understanding linear relationships. It measures how steep a line is and is represented by the letter \(m\) in the slope-intercept form equation \(y = mx + c\).

• The slope is calculated as the ratio of the change in the y-values to the change in the x-values, commonly known as "rise over run."
• A line that rises has a positive slope, while a line that falls has a negative slope.

In this specific exercise, the equation was transformed to \(y = 0.5x - 3\). Here, the slope \(m\) is \(0.5\), indicating that for every unit increase in \(x\), the value of \(y\) increases by 0.5 units. This tells us that the line has a moderate upward tilt.

Y-intercept

The y-intercept is the point where a line crosses the y-axis, which means it occurs where \(x = 0\). In an equation in the slope-intercept form \(y = mx + c\), the y-intercept is represented by \(c\). This point gives a clear, visible starting location for graphing a line on a coordinate plane.

• The y-intercept is very important for graphing because it's an easy first place to plot a line.
• It provides the initial condition of the line before any slope affects it.

In the transformed equation \(y = 0.5x - 3\), the y-intercept \(c\) is \(-3\). This means the line crosses the y-axis 3 units below the origin. By knowing the y-intercept, you can have a concrete point to start drawing your line and from there utilize the slope to complete it.