Problem 75 - Algebra 2

Question

Find the slope of each line. $$ \frac{x}{2}+\frac{y}{3}=1 $$

Step-by-Step Solution

Answer

The slope of the line is \(-\frac{3}{2}\).

Step 1 Title - Rewrite in slope-intercept form

To find the slope of the line represented by the equation, rewrite the equation in the slope-intercept form, which is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Start by isolating \(y\).

Step 2 Title - Isolate y

Multiply both sides of the equation by 6 to clear the fractions: \(6\left(\frac{x}{2}\right) + 6\left(\frac{y}{3}\right) = 6(1)\) resulting in \(3x + 2y = 6\). Now, subtract \(3x\) from both sides to get \(2y = -3x + 6\). Divide both sides by 2 to isolate \(y\): \(y = -\frac{3}{2}x + 3\).

Step 3 Title - Identify the Slope

With the equation now in slope-intercept form \(y = -\frac{3}{2}x + 3\), the coefficient of \(x\), which is \(-\frac{3}{2}\), is the slope of the line.

Key Concepts

Understanding Slope-Intercept Form

Knowing how to work with the slope-intercept form of a line's equation is crucial for studying linear functions. This form is expressed as y = mx + b, where m representss the slope of the line, and b represents the y-intercept – the point where the line crosses the y-axis.

To understand this better, imagine plotting a graph. The slope, m, tells you how steep the line is, while b gives you a starting point on the graph. The beauty of the slope-intercept form is its directness and simplicity, allowing you to sketch a line quickly or interpret the behavior of the function it represents.

When you receive an equation not in this form, such as \(\frac{x}{2} + \frac{y}{3} = 1\), your first task is to convert it to slope-intercept form. This allows you to identify the slope easily and understand how the line will appear on a graph.

Isolating the Y Variable

Isolating the y variable is often the first step in manipulating an algebraic equation into a more usable form, like the slope-intercept form. To isolate y, you perform a series of algebraic operations that 'move' other terms to the opposite side of the equation.

For the given equation \(\frac{x}{2} + \frac{y}{3} = 1\), you start by getting rid of the fractions – a common impediment to having y by itself. By multiplying all terms by the least common denominator, which in our case is 6, you clear the fractions and simplify your calculation work.

After simplification, you then use addition or subtraction to move 'x' terms to the other side of the equation. If y is multiplied by a coefficient, like 2 in the resulting equation 2y = -3x + 6, you finally divide the whole equation by that coefficient to solve for y. This step-by-step process helps you achieve the desired slope-intercept form.

Calculating the Slope

Slope calculation is a fundamental skill in algebra and geometry. The slope represents the rate of change between two variables on the coordinate plane and is commonly denoted as m in equations. In the slope-intercept form y = mx + b, calculating the slope is a matter of identifying the coefficient of x.

From the modified equation y = -\(\frac{3}{2}\)x + 3, the slope m is -\(\frac{3}{2}\). This number tells us that for every unit you move to the right on the x-axis, the value of y decreases by 1.5 units – indicating a downward slope since the slope is negative.

Recognizing the slope not only allows you to understand the direction and steepness of a line but also plays a pivotal role in calculus, physics, and many applied fields. Grasping the concept of slope calculation, therefore, opens the door to deeper understanding and application in various mathematical contexts.