Problem 9 - College Algebra with Corequisite Support

Question

For the following exercises, determine whether the equation of the curve can be written as a linear function. $$ 3 x+5 y=15 $$

Step-by-Step Solution

Answer

Yes, the equation can be written as a linear function: \(y = -\frac{3}{5}x + 3\).

Step 1: Identify the Standard Form

The given equation of the curve is \(3x + 5y = 15\). This is in the standard form of a linear equation \(Ax + By = C\), where \(A = 3\), \(B = 5\), and \(C = 15\).

Step 2: Solve for Y

To write the equation in the form of a linear function \(y = mx + b\), solve for \(y\) in terms of \(x\). Start by isolating \(y\):\[5y = -3x + 15\]

Step 3: Simplify the Equation

Divide all terms by 5 to solve for \(y\):\[y = -\frac{3}{5}x + 3\]

Step 4: Analyze the Result

The equation \(y = -\frac{3}{5}x + 3\) is in slope-intercept form \(y = mx + b\), where the slope \(m = -\frac{3}{5}\) and the y-intercept \(b = 3\).

Key Concepts

Standard Form

Standard form of a linear equation is a neat way to represent a line using three numbers, usually denoted as \(Ax + By = C\). Here, \(A\), \(B\), and \(C\) are constants, and \(x\) and \(y\) are variables representing any point on the line. In our exercise, the equation given was \(3x + 5y = 15\), which fits exactly into this standard form format with:

• \(A = 3\)
• \(B = 5\)
• \(C = 15\)

The beauty of standard form is that it is very straightforward when you want to perform calculations like determining intercepts or testing for parallelism.
It also helps ensure that the equation represents a straight line, making it a linear equation. Understanding how to recognize this form is crucial for identifying linear relationships in math.

Slope-Intercept Form

Another popular way to represent a line in an equation is the slope-intercept form, written as \(y = mx + b\). This form is particularly helpful because it directly shows the slope and the y-intercept of the line.

• \(m\) represents the slope, which tells us how steep the line is. A positive slope means the line rises as it goes from left to right, whereas a negative slope means it falls.
• \(b\) is the y-intercept; this is the point where the line crosses the y-axis.

In the given solution, after solving for \(y\), the equation transforms into \(y = -\frac{3}{5}x + 3\). Here:

• Slope \(m = -\frac{3}{5}\)
• Y-intercept \(b = 3\)

This format is great for graphing a line quickly or understanding the way the line moves across the plane. It's intuitive because you can easily visualize and start plotting the graph right from these two numbers.

Solving for Y

To convert from standard form to slope-intercept form, solving for \(y\) is essential. This involves isolating \(y\) on one side of the equation, so it can be expressed as \(y = mx + b\).
Here's how you can do it step-by-step:First, look at the standard form equation \(3x + 5y = 15\). The goal is to have \(y\) by itself on one side of the equation.

• Subtract \(3x\) from both sides: \(5y = -3x + 15\)
• Divide every term by \(5\) to solve for \(y\): \(y = -\frac{3}{5}x + 3\)

Now, \(y\) is isolated, and the equation is neatly in slope-intercept form.
This process not only simplifies the equation but also makes it incredibly useful for graphing and analyzing linear functions. Solving for \(y\) helps bridge the gap between different forms of linear equations, making it a versatile skill in math.