Problem 56 - Thinking Mathematically

Question

Find the value of \(y\) if the line through the two given points is to have the indicated slope. \((-2, y)\) and \((4,-4), m=\frac{1}{3}\)

Step-by-Step Solution

Answer

The value of \(y\) is 6.

Step 1: Apply the slope formula

The slope \(m\) of a line between two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated by the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substitute the given values into this formula: \(\frac{1}{3} = \frac{-4 - y}{4 - (-2)}\).

Step 2: Simplify the equation

Simplify the equation to isolate \(y\). This results in: \(\frac{1}{3} \times 6 = -4 - y\). Simplify it further to \(2 = -4 - y\).

Step 3: Solve for \(y\)

Rearrange the equation to solve for \(y\). Add 4 on both sides of the equation to isolate \(y\) on one side, resulting in \(2 + 4 = y\), which simplifies to \(y = 6\).

Key Concepts

Slope Formula

To find the slope of a line through two points, you need the slope formula. This formula gives us a way to express the steepness or incline of a line, which is a foundational concept in coordinate geometry.

The slope formula is:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

This formula involves two points on the line, \((x_1, y_1)\) and \((x_2, y_2)\). Here, \(m\) represents the slope, which tells us how much \(y\) changes for a change in \(x\).

For example, in the line passing through points \((-2, y)\) and \((4, -4)\) with a slope \(m = \frac{1}{3}\), we substitute these values into the formula to set up the equation:

• \( \frac{1}{3} = \frac{-4 - y}{4 - (-2)} \)

The denominator (\(x_2 - x_1\)) evaluates to 6, providing a crucial component as we solve for \(y\).

Solving Linear Equations

Once the slope formula is set up, we transform it into a linear equation to solve for the unknown variable. Linear equations often involve simple algebraic manipulations to isolate the desired variable.

Starting with the equation derived from the slope formula:

• \( \frac{1}{3} = \frac{-4 - y}{6} \)

Cross-multiply to eliminate the fraction, resulting in a simpler equation to manage. With the fraction step resolved, we get:

• \( 2 = -4 - y \)

This equation means we need to add or subtract numbers to both sides to solve for \(y\).

The next step is to isolate \(y\) by adding 4 to both sides, to rearrange the equation:

• \( 2 + 4 = y \)
• \( y = 6 \)

Through these steps, you can systematically solve linear equations for any unknown.

Point-Slope Form

The point-slope form is a useful equation of a line, particularly when a point on the line and the slope are known. This form helps visualize how a line behaves and is derived from the slope formula itself.

The point-slope form is written as:

• \( y - y_1 = m(x - x_1) \)

Where \( (x_1, y_1) \) is a point on the line, and \( m \) is the slope. This equation gives us a linear framework to understand lines in geometry.

For instance, if you were working with the point \((4, -4)\) and slope \(\frac{1}{3}\), the equation would initiate as:

• \( y + 4 = \frac{1}{3}(x - 4) \)

This method directly connects the slope and a specific point on the line to form an equation. It's particularly handy when needing to derive further properties of a line or when working with different forms of a linear equation.