Problem 39 - Algebra 1

Question

Write an equation of the line that is parallel to the given line and passes through the given point. $$y=6 x+9,(5,-3)$$

Step-by-Step Solution

Answer

The equation of the line that is parallel to \(y = 6x + 9\) and passes through the point (5,-3) is \(y = 6x - 33\).

Step 1: Identifying and Setting up the Equation

Start by writing the equation of the line in y = mx + b form. Since we know that parallel lines have the same slope, the slope m of the new line will be the same as the slope of the given line, which is 6. Write the equation of the new line as \(y = 6x + b\), where b is the y-intercept we need to find.

Step 2: Finding the y-intercept

To find b, substitute the point (5, -3) into the equation. This point lies on the line, so it must satisfy the line equation. Therefore, the coordinates x = 5 and y = -3 can replace x and y in the equation, so it becomes \(-3 = 6(5) + b\).

Step 3: Solve for b

Solve the equation \(-3 = 30 + b\), rearrange for b to obtain \(b = -3 - 30\). Therefore, b = -33.

Step 4: Write Final Equation

The final step is to write down the equation of the line with the values of m (slope) and b (y-intercept) determined. The equation of the line is therefore \(y = 6x - 33\).

Key Concepts

Slope-Intercept Form

The slope-intercept form is one of the most common ways to express a linear equation. It allows you to quickly identify the slope and the y-intercept of a line. The general format for this form is \(y = mx + b\), where:

• \(y\) is the dependent variable (usually representing the vertical axis).
• \(x\) is the independent variable (usually representing the horizontal axis).
• \(m\) is the slope of the line, which represents the change in \(y\) for a unit change in \(x\).
• \(b\) is the y-intercept, which is the value of \(y\) when \(x\) is zero.

In this form, the slope \(m\) shows how steep the line is, and the y-intercept \(b\) tells us where the line crosses the y-axis.
Parallel lines have the same slope, so when writing an equation for a line parallel to another, maintain the same slope value from the original line equation.

Finding Y-Intercept

Finding the y-intercept of a line involves determining the value at which the line crosses the y-axis. This occurs at \(x = 0\), but often, when dealing with parallel lines, you're given a point and a slope instead.
To find the y-intercept when given a point \((x_1, y_1)\) on the line and the slope \(m\), you can use the slope-intercept form by plugging in the point into the equation \(y = mx + b\). Here’s a step-by-step:

• Substitute \(x_1\) and \(y_1\) into the equation, replacing \(x\) and \(y\) respectively. This gives the equation \(y_1 = mx_1 + b\).
• Solve the equation for \(b\) to find the y-intercept.
• This requires simple algebra, where you isolate \(b\) on one side of the equation.

In the original exercise, substituting the point \((5, -3)\) into the equation \(y = 6x + b\) leads us to the equation \(-3 = 6(5) + b\). Solving this provides the y-intercept \(b = -33\).

Linear Equations

Linear equations form straight lines when graphed on a coordinate plane. They are algebraic equations of the first degree in two variables and can be represented in several forms.
The most straightforward form is the slope-intercept form (\(y = mx + b\)), but they can also be expressed in standard form (\(Ax + By = C\)) or point-slope form \(y - y_1 = m(x - x_1)\).

• Linear equations have no exponents higher than one, meaning they don't curve or form a bend like quadratic equations might.
• Solutions to linear equations are simple straight lines with constant slope \(m\).
• Understanding linear equations is crucial for solving problems involving parallelism and intersection, like the one in the given exercise.

In the context of the exercise, the task was to find a linear equation parallel to a given line. This emphasizes the importance of linear relationships and consistent slopes when considering parallel lines.