Problem 49 - Algebra 1

Question

Write an equation of the line that passes through the points. (5,1),(3,-6)

Step-by-Step Solution

Answer

The equation of the line that passes through the points (5,1) and (3,-6) is \(y = 3.5x - 7.5\)

Step 1: Calculate the slope

To calculate the slope \(m\) of the line, use the formula \(m = (y_2 - y_1) / (x_2 - x_1)\), substituting the given points (5,1) and (3,-6) into the formula. This gives: \(m = (-6 - 1) / (3 - 5) = -7 / -2 = 3.5\).

Step 2: Insert the slope and a point into the line equation

Next, substitute the slope and a point into the line equation. The slope is \(m = 3.5\) and the y-intercept \((x,y)\) can be any of the given points, so use (5,1). This gives: \(1 = 3.5 * 5 + c\).

Step 3: Solve for y-intercept

Solving the above equation for \(c\), the y-intercept, gives \(c = 1 - 3.5 * 5 = -7.5\).

Key Concepts

Slope of a Line

The slope of a line is a measure of how steep the line is. It indicates the rate at which the line rises or falls as you move along it. To find the slope, you can use the formula:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

Here, \((x_1, y_1)\) and \((x_2, y_2)\) are two points on the line. The difference in the y-values is divided by the difference in the x-values. This will yield the slope \(m\).
For example, in the line passing through points \((5, 1)\) and \((3, -6)\), the slope is calculated as:
\( m = \frac{-6 - 1}{3 - 5} = \frac{-7}{-2} = 3.5 \)
This positive slope of 3.5 means that for every two units you move to the right on the x-axis, the line rises 3.5 units.

Point-Slope Form

The point-slope form is a useful way to write the equation of a line when you know one point on the line and its slope. The formula for point-slope form is:

• \( y - y_1 = m(x - x_1) \)

Here, \((x_1, y_1)\) represents a specific point on the line and \(m\) is the line's slope.
Using this forms makes it easier to construct the full equation of the line. For example, if we know the slope is 3.5 and the line passes through the point \((5, 1)\), we can substitute these into the point-slope form:
\( y - 1 = 3.5(x - 5) \)
This equation can now be simplified to find other useful forms of the line's equation, such as the slope-intercept form.

Y-Intercept

The y-intercept of a line is the point where the line crosses the y-axis. It is represented by the coordinate \((0, c)\). This is an important concept as it indicates the value of \(y\) when \(x = 0\).
To find the y-intercept, you can substitute the slope and one point on the line into the linear equation form:

• \( y = mx + c \)

In our example, we've already calculated the slope \(m = 3.5\) and have used the point \((5, 1)\). Substituting these into the equation:
\( 1 = 3.5 \times 5 + c \)
Solving for \(c\), we get \(c = -7.5\). Therefore, the equation of the line in slope-intercept form becomes \( y = 3.5x - 7.5 \).
This means the line crosses the y-axis at \(-7.5\).