Problem 21 - Precalculus Mathematics for Calculus

Question

Find an equation of the line that satisfies the given conditions. Through \((1,7) ; \quad\) slope \(\frac{2}{3}\)

Step-by-Step Solution

Answer

The equation is \( y = \frac{2}{3}x + \frac{19}{3} \).

Step 1: Understand the Point-Slope Formula

To find the equation of a line, we can use the point-slope form of a line's equation, which is \( y - y_1 = m(x - x_1) \). Here, \(m\) is the slope, and \((x_1, y_1)\) is a point on the line. We have all necessary information: the slope \( \frac{2}{3} \) and the point \((1, 7)\).

Step 2: Substitute Known Values

Substitute the known values into the point-slope formula. The point is \((1, 7)\) and the slope \(m = \frac{2}{3}\).\[ y - 7 = \frac{2}{3}(x - 1) \]

Step 3: Simplify the Equation

Distribute the slope \( \frac{2}{3} \) in the equation.\[ y - 7 = \frac{2}{3}x - \frac{2}{3} \]

Step 4: Solve for y to Get the Slope-Intercept Form

Add 7 to both sides to solve for \(y\) and convert the equation to the slope-intercept form, \(y = mx + b\).\[ y = \frac{2}{3}x - \frac{2}{3} + 7 \] \[ y = \frac{2}{3}x + \frac{21}{3} - \frac{2}{3} \] \[ y = \frac{2}{3}x + \frac{19}{3} \]

Step 5: Write the Final Equation

Thus, the equation of the line in slope-intercept form is:\[ y = \frac{2}{3}x + \frac{19}{3} \]

Key Concepts

Point-Slope Form

The point-slope form is a handy way to write the equation of a line when you know one point on the line and the slope of the line. It is expressed as:

• \( y - y_1 = m(x - x_1) \)

In this equation:

• \(m\) is the slope of the line.
• \((x_1, y_1)\) is a point on the line.

For example, in the exercise where we have a point \((1, 7)\) and a slope \(\frac{2}{3}\), we substitute these values into the formula:

• \( y - 7 = \frac{2}{3}(x - 1) \)

This form is particularly advantageous because it makes it very simple to plug in the values and obtain the line's equation quickly. It focuses on finding the change in \(y\) for the change in \(x\), which is essentially what the slope tells us. Think of it as positioning a string (the slope) from a specific anchor point \((x_1, y_1)\) stretching across a graph.

Slope-Intercept Form

Once you have the equation in point-slope form, transforming it to the slope-intercept form makes it easy to graph and interpret. The slope-intercept form is given by:

• \( y = mx + b \)

Here:

• \(m\) is the slope, indicating the rate of change.
• \(b\) is the y-intercept, where the line crosses the y-axis.

In our sample problem, we started with:

• \( y - 7 = \frac{2}{3}(x - 1) \)

To convert, solve for \(y\) by first distributing the slope through the expression:

• \( y - 7 = \frac{2}{3}x - \frac{2}{3} \)
• Then isolate \(y\) by adding 7 to each side:

• \( y = \frac{2}{3}x + \frac{21}{3} - \frac{2}{3} \)

This results in:

• \( y = \frac{2}{3}x + \frac{19}{3} \)

Now you've got a clear visual layout—where the slope \( \frac{2}{3} \) dictates the tilt, and \( \frac{19}{3} \) tells you where it intersects the y-axis. This reveals how easily you can graph the line.

Linear Equations

Linear equations are expressions that construct straight lines when graphed on a coordinate plane. Such equations reveal crucial relationships between two variables. The basic form is captured in:

• \( y = mx + b \)

Whether using point-slope or slope-intercept forms, linear equations are adept at representing real-world phenomena that scale predictably. For example, recognizing that the equation:

• \( y = \frac{2}{3}x + \frac{19}{3} \)

Means that for every change of 3 units along the x-axis, the y-value will expand by 2. The straightforward nature of linear equations is essential for modeling direct relationships where there is a simple proportional dependency between variables. Additionally, linear equations are foundational knowledge for more advanced mathematics, opening the door to calculus and beyond. They are pivotal in understanding how quantities vary together, and with their predictability, can be used in numerous practical applications, from calculating distances over time to predicting costs.