Problem 28 - College Algebra

Question

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 is \(y = 2x - 5\).

Step 1: Understanding the problem

We need to find the equation of a line that is parallel to the line given by the equation \(y = 2x + 5\) and passes through the point \((4, 3)\). A parallel line will have the same slope as the given line.

Step 2: Identifying the slope of the given line

The given line equation is in slope-intercept form \(y = mx + b\), where \(m\) is the slope. Here, the slope \(m\) is 2. Since the lines are parallel, the new line will also have a slope of 2.

Step 3: Using the point-slope formula

The point-slope formula is \( y - y_1 = m(x - x_1) \), where \(m\) is the slope and \((x_1, y_1)\) is the point the line passes through. Plugging in the slope \(m = 2\) and the point \((4, 3)\), we get: \[ y - 3 = 2(x - 4) \]

Step 4: Solving the point-slope equation

Distribute the slope on the right side: \[ y - 3 = 2x - 8 \] Then solve for \(y\) by adding 3 to both sides to convert to slope-intercept form: \[ y = 2x - 8 + 3 \] \[ y = 2x - 5 \]

Step 5: Convert to the slope-intercept form

The equation is now completely in slope-intercept form, which is \(y = mx + b\). Here, the equation of the new line is \(y = 2x - 5\). It has a slope of 2 and a y-intercept of -5.

Key Concepts

Point-Slope Formula

The point-slope formula is a handy tool for writing the equation of a line when you know the slope and a point it passes through. This formula is given by \( y - y_1 = m(x - x_1) \). Here:

• \( m \) is the slope of the line,
• \((x_1, y_1)\) is a specific point on the line.

This formula is particularly useful when the slope and a point are known but the y-intercept isn't immediately available. By rearranging this formula, you can find the line's equation in the slope-intercept form easily. Simply solve for \( y \) and you'll have a neat and clean equation that describes your line.

Slope-Intercept Form

The slope-intercept form of a line is written as \( y = mx + b \), where:

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

This form is extremely popular because it's straightforward and immediately tells you both the slope and the y-intercept. Once we have the equation in point-slope form, we can simply solve for \( y \) to convert it to this slope-intercept form. This makes it easier to graph the line and understand its orientation in relation to the axes.

Parallel Lines

Parallel lines are lines in a plane that never meet. They have the same slope but different y-intercepts. When two lines are parallel, their steepness is identical.

• Their equations will differ only by the y-intercept.
• In our example, the line \( y = 2x + 5 \) has a slope of 2.
• A parallel line passing through the point (4,3) would also have a slope of 2.

Understanding parallel lines helps in predicting where they will be and knowing that they maintain the same distance apart. Parallel lines are very common in geometry and real-world applications where consistent spacing is vital.