Problem 58 - Algebra 1

Question

What is the equation of the line that passes through the points \((7,4)\) and \((-5,-2) ?\) (A) \(y=\frac{1}{2} x-\frac{1}{2}\) (B) \(y=-\frac{1}{2} x+\frac{1}{2}\) (c) \(y=-\frac{1}{2} x-\frac{1}{2}\) (D) \(y=\frac{1}{2} x+\frac{1}{2}\)

Step-by-Step Solution

Answer

The correct answer is (A) \(y = 1/2x + 1/2\).

Step 1: Calculate the Slope of the Line

First, let's calculate the slope \(m\) of the line. We use the formula \(m = (y_2 - y_1) / (x_2 - x_1)\). Substituting the given points into this formula, we have \(m= (4-(-2) )/( 7-(-5) )\). This simplifies to \(m=6/12 = 0.5 = 1/2\). So the slope of the line is \(1/2\).

Step 2: Calculate the Y-intercept

Now let's solve for the y-intercept \(c\) by substituting one of the given points and the calculated slope into the equation of the line. Let's use the point (7, 4) and the slope \(1/2\) into the formula \(y = mx + c\). When we substitute, we get \(4 = 1/2 * 7 + c\). Solving for \(c\), we get \(c = 4 - 7/2 = 0.5 = 1/2\). So the y-intercept is \(1/2\).

Step 3: Write the equation of the line

The formula for a line is \(y = mx + c\). Now that we know the slope \(m = 1/2\) and y-intercept \(c = 1/2\), we can write the equation as \(y = 1/2 * x + 1/2\) which is the same as \(y = 1/2x + 1/2\).

Key Concepts

Slope Calculation

To find the equation of a line, the slope calculation is an essential first step. The slope of a line, often denoted by the symbol \( m \), measures the steepness and direction of the line. The formula to calculate the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:\[m = \frac{y_2 - y_1}{x_2 - x_1}\,\]which represents the "rise" over the "run".
Here, the rise is the change in the \( y \)-values, and the run is the change in the \( x \)-values.
Let's apply this formula using the points provided: \((7,4)\) and \((-5,-2)\).

• Calculate the change in \( y \): \( 4 - (-2) = 6 \)
• Calculate the change in \( x \): \( 7 - (-5) = 12 \)

Substitute these into the formula:\[m = \frac{6}{12} = \frac{1}{2}\,\]This means the slope \( m \) of the line is \( \frac{1}{2} \). A positive slope indicates the line rises from left to right.

Y-Intercept

Once the slope of a line is determined, the next step is to find the y-intercept. The y-intercept is the point where the line crosses the y-axis, described by the equation \( y = mx + c \) where \( c \) is the y-intercept.
The y-intercept is crucial because it allows you to graph the line accurately.
To calculate \( c \), substitute one of the points (we'll use \((7, 4)\)) and the slope \( \frac{1}{2} \) into the line equation:\[4 = \frac{1}{2} \cdot 7 + c\,\]Solve for \( c \):

• \( 4 = \frac{7}{2} + c \)
• Subtract \( \frac{7}{2} \) from both sides: \( c = 4 - \frac{7}{2} \)
• Simplify: \( c = \frac{8}{2} - \frac{7}{2} = \frac{1}{2} \)

Therefore, the y-intercept \( c \) is \( \frac{1}{2} \). This informs us that our line crosses the y-axis at \((0,\frac{1}{2})\).

Point-Slope Formula

Sometimes, the point-slope formula comes in handy when writing the equation of a line. This formula is especially useful when you know the slope of the line and a point on it. The point-slope form of a line’s equation is:\[y - y_1 = m(x - x_1)\,\]where \((x_1, y_1)\) is a point on the line and \(m\) is the slope.
This format quickly allows you to build an equation when dealing with specific point-slope scenarios.
Using our example with the point \((7,4)\) and slope \(\frac{1}{2}\), substitute into the point-slope formula:\[y - 4 = \frac{1}{2}(x - 7)\,\]Simplifying this:

• Distribute the slope: \( y - 4 = \frac{1}{2}x - \frac{7}{2} \)
• Add 4 to both sides to isolate \( y \): \( y = \frac{1}{2}x - \frac{7}{2} + 4 \)
• Simplify further: \( y = \frac{1}{2}x + \frac{1}{2} \)

And there you have it—the line’s equation matches our earlier form. The point-slope formula is a powerful tool for quickly identifying the relationship between constants and variables in linear functions.