Problem 52 - College Algebra

Question

For the following exercises, write an equation for the line described. Write an equation for a line parallel to \(f(x)=-5 x-3\) and passing through the point (2,-12) .

Step-by-Step Solution

Answer

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

Step 1: Identify the Slope of the Given Line

The first step to solving this problem is to identify the slope of the line described by the equation \( f(x) = -5x - 3 \). The slope, \( m \), can be found directly from the equation, as it is the coefficient of \( x \). Thus, the slope of the line is \( m = -5 \).

Step 2: Identify the Slope of the Parallel Line

By definition, parallel lines have the same slope. Therefore, the line we are looking to find will also have a slope of \(-5\).

Step 3: Use the Point-Slope Form

We know the slope of our new line, and we have been given a point \((2, -12)\) that it passes through. We can use the point-slope form of a linear equation, which is \( y - y_1 = m(x - x_1) \), where \((x_1, y_1)\) is a point on the line and \(m\) is the slope.

Step 4: Substitute Given Values into the Point-Slope Form

Substitute \(m = -5\), \( x_1 = 2 \), and \( y_1 = -12 \) into the equation: \( y - (-12) = -5(x - 2) \).

Step 5: Simplify the Equation

Expand and simplify the equation: \( y + 12 = -5x + 10 \).

Step 6: Solve for y to Write in Slope-Intercept Form

Isolate \( y \) to get the equation into slope-intercept form (\( y = mx + b \)): \( y = -5x + 10 - 12 \).

Step 7: Finalize the Equation

Combine constants to finalize the equation: \( y = -5x - 2 \). This is the equation of the line parallel to \( f(x) = -5x - 3 \) that passes through \( (2, -12) \).

Key Concepts

Slope-Intercept Form

One of the most commonly used ways to express a linear equation is the slope-intercept form. It is written as \( y = mx + b \), where \( m \) represents the slope of the line, indicating how steep the line is, and \( b \) is the y-intercept, the point where the line crosses the y-axis. This form is highly practical because it provides immediate information about the line:

• The slope \( m \) tells us the rise over run—meaning, how many units the line goes up or down for every unit it goes across.
• The y-intercept \( b \) lets us know where the line starts on the y-axis when \( x \) is zero.

One can easily sketch a graph of a linear equation by identifying these two components. This form is particularly favorable when you need to graph the line or quickly understand how the line behaves.

Point-Slope Form

When you know a point a line passes through and its slope, the point-slope form comes handy. It looks like this: \( y - y_1 = m(x - x_1) \). Here, \( (x_1, y_1) \) is a coordinate point that the line intersects, and \( m \) is the slope of the line.

• The point-slope form is especially useful for writing the equation of a line when you're given one point and the slope, without needing to know where it crosses the y-axis.
• By substituting the known values into the formula, you can derive the equation of the line.

In our original exercise, we used this form to create the equation of a line parallel to another, passing through a specific point. Simply plug in the slope of \(-5\) and the point \((2, -12)\) to get the desired equation.

Linear Equations

Linear equations are foundational in algebra and represent relationships between variables. They generally describe a straight line when graphed. The canonical form of a linear equation is \( ax + by = c \), but it's often transformed into other forms, like the slope-intercept or point-slope form, to solve specific problems.

• They consist of variables which are raised to the power of one, making their graphs straight lines.
• The coefficients tell us the slope and intercept, helping depict the direction and position of the line.

Understanding linear equations helps in recognizing patterns and changes in different fields, such as physics or economics. They serve as stepping stones to more complex algebraic concepts, like systems of equations, and are crucial for solving real-world problems.