Problem 27 - College Algebra

Question

Find the equation of the line using the point-slope formula. Write all the final equations using the slope-intercept form. \((1,3)\) and \((5,5)\)

Step-by-Step Solution

Answer

The equation of the line is \(y = \frac{1}{2}x + \frac{5}{2}\).

Step 1: Find the slope

To find the slope of the line that passes through the points \((1,3)\) and \((5,5)\), use the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Substituting the values, we get:\[ m = \frac{5 - 3}{5 - 1} = \frac{2}{4} = \frac{1}{2} \]So, the slope \(m\) is \(\frac{1}{2}\).

Step 2: Use the point-slope form

The point-slope form of a line equation is:\[ y - y_1 = m(x - x_1) \]Using the point \((1,3)\) and the slope \(m = \frac{1}{2}\), we substitute these into the formula:\[ y - 3 = \frac{1}{2}(x - 1) \]

Step 3: Simplify to slope-intercept form

We will simplify the equation from the point-slope form to the slope-intercept form \(y = mx + b\).Start with the equation: \[ y - 3 = \frac{1}{2}(x - 1) \]Distribute \(\frac{1}{2}\) to \((x - 1)\):\[ y - 3 = \frac{1}{2}x - \frac{1}{2} \]Add 3 to both sides:\[ y = \frac{1}{2}x + \frac{5}{2} \]This is the equation of the line in slope-intercept form.

Key Concepts

Slope-Intercept Form

The slope-intercept form of an equation of a line is a widely used format that allows us to easily identify the slope and y-intercept of the line. It is typically written as:\[ y = mx + b \]

• Here, \(m\) represents the slope of the line.
• \(b\) is the y-intercept, which is the point where the line crosses the y-axis.

This form is useful because once you know the slope and y-intercept, you can quickly graph the line or understand how the line behaves.
For example, in the equation \(y = \frac{1}{2}x + \frac{5}{2}\), the line has a slope of \(\frac{1}{2}\) and crosses the y-axis at \(\frac{5}{2}\). This tells us that as we move right on the x-axis, the line rises by half a unit for every full unit across.

Point-Slope Formula

The point-slope formula is another essential form used to find the equation of a line. Unlike the slope-intercept form, this form is particularly useful when you know a point on the line and the slope and is expressed as:\[ y - y_1 = m(x - x_1) \]

• \((x_1, y_1)\) represents any point on the line.
• \(m\) is the slope.

This format is helpful because it allows us to write the equation of a line effortlessly when we already have one specific point and the slope. We used this in our original problem by plugging in the point \((1,3)\) and slope \(m = \frac{1}{2}\) to get:\[ y - 3 = \frac{1}{2}(x - 1) \]From there, we can rearrange into the slope-intercept form to make it even more useful for graphing or analyzing the line.

Slope Calculation

Calculating the slope is a fundamental skill when dealing with linear equations. The slope indicates how steep a line is and is calculated using any two points on the line. The formula used for calculating slope is:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

• \((x_1, y_1)\) and \((x_2, y_2)\) are two points on the line.
• The subtraction \(y_2 - y_1\) finds the difference in the y-values between the two points (rise).
• The subtraction \(x_2 - x_1\) finds the difference in the x-values (run).

In the exercise, we used the points \((1,3)\) and \((5,5)\). Plugging these into the formula, we have:\[ m = \frac{5 - 3}{5 - 1} = \frac{2}{4} = \frac{1}{2} \]This calculation shows us that for every 2 units of vertical change, there is a 4 unit change horizontally, simplifying to a slope of \(\frac{1}{2}\). This means the line rises one unit for every two units it moves to the right.