Problem 28 - College Algebra with Corequisite Support

Question

For the following exercises, find the equation of the line using the point- slope formula. Write all the final equations using the slope-intercept form. parallel to \(y=2 x+5\) and passes through the point (4,3)

Step-by-Step Solution

Answer

The equation of the line is \(y = 2x - 5\).

Step 1: Identify the Given Slope

The problem states that the line is parallel to the line given by the equation \(y = 2x + 5\). Since parallel lines have the same slope, the slope \(m\) of our line is \(2\).

Step 2: Use Point-Slope Formula

With the slope \(m = 2\) and the point \((4, 3)\), use the point-slope formula which is \(y - y_1 = m(x - x_1)\). Substitute the values into the equation: \(y - 3 = 2(x - 4)\).

Step 3: Simplify the Equation

Distribute the 2 on the right side of the equation: \(y - 3 = 2x - 8\).

Step 4: Convert to Slope-Intercept Form

Add 3 to both sides of the equation to solve for \(y\): \(y = 2x - 5\). Now, the equation is in slope-intercept form \(y = mx + b\).

Key Concepts

Slope-Intercept Form

The slope-intercept form of a line is a key concept in algebra. It is a way of writing the equation of a line so that the slope and the y-intercept are immediately apparent. It takes the form \(y = mx + b\), where \(m\) is the slope of the line, and \(b\) is the y-intercept.
To solve the given problem, we needed to write the final equation in slope-intercept form. This format enables you to quickly identify how steep the line is (the slope) and where it crosses the y-axis (the y-intercept).
Some advantages of using the slope-intercept form include:

• It makes it easy to graph the line.
• Calculating and interpreting the slope and intercept is straightforward.

The problem starts by using the point-slope form, a common technique when given a point and a slope, and then transitions to the slope-intercept form for clarity and simplicity.

Parallel Lines

Parallel lines are an interesting aspect of geometry and algebra. Two lines are parallel if they have the same slope and will never intersect. In the problem, the given line \(y = 2x + 5\) sets the slope for our new line because parallel lines must have the same slope.
This means our new line also has a slope \(m = 2\). Understanding this relationship is crucial because it allows us to establish a new line with a simple substitution of the slope.
Here's why parallel lines are important:

• They help in constructing and analyzing geometric shapes, like rectangles and parallelograms.
• Knowing about parallel lines aids in real-world problem-solving, for example, in architecture and engineering.

Keeping the same slope maintains the parallel orientation between the two lines, which is fundamental for many mathematical applications.

Equation of a Line

The equation of a line is an algebraic expression that represents all the points along the line. Having the equation allows you to graph the line and interpret its characteristics. There are several forms to write this, but the point-slope form and slope-intercept form are among the most useful.
In the exercise, we utilized the point-slope formula \(y - y_1 = m(x - x_1)\) to find the equation of the line that is parallel to \(y = 2x + 5\) and passes through the point \((4, 3)\). This first step involved finding the equation using the given point and slope.
By utilizing:

• The point-slope form, which provides a straightforward way to use known points on the line.
• The slope-intercept form, offering an easy conversion to interpret the line's characteristics.

Mastering the equation of a line is essential, whether you're solving math problems or tackling real-world tasks that involve calculations of gradients and projections.