Problem 40 - Algebra 1

Question

Write an equation of the line that is parallel to the given line and passes through the given point. $$y=-9 x-8,(7,-2)$$

Step-by-Step Solution

Answer

The equation of the line is \(y = -9x + 61\).

Step 1: Identify the slope of the given line

From the equation \(y=-9x -8\), it can be seen that the slope (m) of the given line is -9.

Step 2: Use the point-slope form of the line equation

Substitute the slope (-9) and the given point \((7,-2)\) into the point-slope form \(y - y_{1} = m(x - x_{1})\). This gives \(y - (-2) = -9(x - 7)\).

Step 3: Simplify the equation

Simplify the equation gained from step 2: \(y + 2 = -9x + 63\), which can be further simplified to \(y = -9x + 61\) by subtracting 2 from both sides of the equation. This is the equation of the required line.

Key Concepts

Understanding Slope

The slope of a line, often denoted by the letter "m," is a measure of how steep the line is. It describes the rate at which the line rises or falls. In mathematical terms, it is the change in the "y" values divided by the change in the "x" values between two points.

• If a line goes upwards as you move from left to right, it has a positive slope.
• If the line goes downwards, the slope is negative.
• For perfectly horizontal lines, the slope is zero, while vertical lines have an undefined slope.

In our exercise, the original line equation is given as \(y = -9x - 8\). Here, the slope is \(-9\), indicating a steep downward slope. By knowing this, we ensure that any new line drawn parallel to it will also have the same slope.

The Point-Slope Form

The point-slope form of a line equation is a powerful tool in coordinate geometry. It provides a simple way to write the equation of a line when you know the slope and a single point on the line. The formula is given by:\[ y - y_1 = m(x - x_1) \]Where \(m\) represents the slope of the line, and \((x_1, y_1)\) represents a specific point the line passes through. For our exercise, the known slope is \(-9\) and the point through which the line passes is \((7, -2)\). Substituting these values into the formula, we have:\[ y - (-2) = -9(x - 7) \]This setup is crucial as it perfectly links a geometric concept (the slope) with algebraic expressions (line equations).

Formulating the Line Equation

The line equation can be uncovered through the point-slope form as a starting point. By substituting the values from our exercise into the point-slope formula, the intermediate equation is:\[ y + 2 = -9(x - 7) \]This form relies on the specific information about the point the line passes through and its slope. It is quite flexible as it can be directly converted into other common forms of a line equation like the slope-intercept form, where an equation is shaped as \(y = mx + b\). Transitioning from the point-slope form to a typical line equation format involves expanding the equation and performing algebraic manipulations to isolate \(y\) on one side.

Steps to Simplification

Simplification is about making algebraic expressions as clear and straightforward as possible. For line equations, it usually means solving for \(y\):From the equation \(y + 2 = -9x + 63\), we begin the process by performing operations that lead to smaller coefficients and an easier-to-read formula. Follow these steps:

• Subtract 2 from both sides.
• Reorder terms to perfectly fit the \(y = mx + b\) format.

Hence, the simplified equation of our line becomes:\[ y = -9x + 61 \]This expression now clearly shows the slope of the line and the y-intercept, making it easier to plot and understand in the Cartesian coordinate plane.