Problem 26 - College Algebra

Question

Find the equation of the line using the point-slope formula. Write all the final equations using the slope-intercept form. \((-3,10)\) and \((5,-6)\)

Step-by-Step Solution

Answer

The equation of the line is \( y = -2x + 4 \).

Step 1: Calculate the Slope

To find the equation of a line, we first need the slope. The slope (m) between two points \(x_1, y_1\) and \(x_2, y_2\) is calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Substitute \(x_1 = -3, y_1 = 10\), and \(x_2 = 5, y_2 = -6\):\[ m = \frac{-6 - 10}{5 - (-3)} = \frac{-16}{8} = -2 \]

Step 2: Use Point-Slope Formula

With the slope \(m = -2\) known, we can write the equation of the line using the point-slope formula, \(y - y_1 = m(x - x_1)\).Choose one of the points, say \((-3, 10)\):\[ y - 10 = -2(x + 3) \]

Step 3: Simplify to Slope-Intercept Form

Next, simplify the equation from the point-slope form to the slope-intercept form, \(y = mx + b\).Start with \(y - 10 = -2(x + 3)\):Expand the equation: \[ y - 10 = -2x - 6 \]Add 10 to both sides to solve for \(y\):\[ y = -2x - 6 + 10 \]Simplify:\[ y = -2x + 4 \]

Key Concepts

Slope-Intercept Form

The slope-intercept form is a popular way to express the equation of a line. It succinctly provides the slope and the y-intercept, making graphing easier. This form is defined as follows:
\[ y = mx + b \]
where:

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

This form is particularly useful because it clearly shows the gradient of the line and its initial value when \( x = 0 \). Once you have the equation in this format, you can quickly determine how the line behaves and how it will appear on a coordinate plane.

Equation of a Line

An equation of a line in mathematics provides a way to express a straight line using algebraic expressions. There are different forms, but two main ones are the slope-intercept form and the point-slope form.

• *Slope-Intercept Form*: This form, \( y = mx + b \), is excellent for quickly identifying the slope and the y-intercept.
• *Point-Slope Form*: This one is useful when you know the slope of a line and a point on the line. It is given as \( y - y_1 = m(x - x_1) \).

Using these forms, you can describe any straight line in a two-dimensional space. Transforming between these forms often involves algebraic manipulation, enabling you to switch perspectives depending on the information you have. Knowing these forms allows for flexibility in solving problems related to linear equations.

Slope Calculation

Calculating the slope is an essential step in determining the equation of a line. The slope, denoted by \( m \), expresses how steep the line is and in which direction it is inclined. To calculate the slope between two points, use the simple formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Here, the numerators represent the change in the \( y \)-value, while denominators represent the change in the \( x \)-value between the two points. Using this formula, you can measure how much \( y \) changes for each unit of \( x \) change.
In the original exercise, this formula was applied to the points \((-3,10)\) and \((5,-6)\) yielding a slope of \(-2\). This negative value indicated the line slants downwards as it moves from left to right. Slope calculation is vital as it helps to set the building blocks for constructing the rest of the line's equation.