Problem 22 - Elementary Algebra

Question

Find \(y\) if the line through the points \((12,14)\) and \((3, y)\) has a slope of \(\frac{4}{3}\).

Step-by-Step Solution

Answer

The value of \( y \) is 2.

Step 1: Understand the Slope Formula

The slope of a line that passes through two points \(x_1, y_1\) and \(x_2, y_2\) is calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]We need to find the value of \(y\) that makes the slope of the line through the points \( (12,14) \) and \( (3,y) \) equal to \frac{4}{3}\.

Step 2: Set Up the Slope Equation

Plug the points \( (12,14) \) and \( (3,y) \) into the slope formula:\[ \frac{y - 14}{3 - 12} = \frac{4}{3} \]Simplify the denominator, observing that it is just the difference in the \(x\)-coordinates.

Step 3: Simplify the Denominator

Calculate \(3 - 12\) to get \(-9\). Now the slope equation is:\[ \frac{y - 14}{-9} = \frac{4}{3} \].

Step 4: Solve for y

To solve for \(y\), cross-multiply the fractions:\[ 3(y - 14) = -9 \times 4 \]This gives:\[ 3(y - 14) = -36 \].

Step 5: Distribute and Isolate y

Distribute the \(3\) on the left side:\[ 3y - 42 = -36 \]To isolate \(y\), add \(42\) to both sides:\[ 3y = 6 \].

Step 6: Find the Value of y

Finally, divide both sides by \(3\) to solve for \(y\):\[ y = 2 \].

Key Concepts

Slope Formula

The slope formula is an essential concept in algebra which helps us determine the steepness or incline of a line on a graph. Imagine the surface of a hill. The steeper it is, the higher its slope. In coordinate geometry, we calculate this slope using two points on a line. If you have two points, \((x_1, y_1)\) and \((x_2, y_2)\), you can find the slope \(m\) with the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]The numerator represents the change in the \(y\)-values, which is also called the rise. The denominator shows the change in the \(x\)-values, often referred to as the run. By using this formula, you get an idea of how much the line goes up or down as it moves from left to right.

Solving Linear Equations

Solving linear equations involves finding the value of a variable that makes the equation true. In the given problem, we need to determine the value of \(y\) that fits the slope condition. Here is a structured way to solve:

• Substitute the given points and the slope into the slope formula.
• Simplify the equation by performing necessary arithmetic operations like subtraction or simplification.
• Cross-multiply if fractions are involved to eliminate the denominators.

From these steps, you'll end up with a simpler equation. Sometimes, like in the example, cross-multiplication helps you move closer to isolating the variable. Distributing and isolating the variable are common steps. Let's look at it in action: First, cross-multiplying the fractions \( \frac{y - 14}{-9} = \frac{4}{3} \) provides a straightforward path to isolating \(y\). Keep a keen eye on doing equal operations on both sides, like adding or subtracting terms, to ensure the balance of the equation.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, bridges algebra with geometry. In this context, it's the study of lines through points on a plane. Points on a plane are identified by their coordinates—like addresses, they tell us the exact location.Coordinate geometry allows us to explore various geometric questions using algebraic methods.When dealing with lines,

• you often start with pinpointing two specific points on the line.
• Using the points, you can utilize the slope formula to find how steep the line is.
• This understanding of lines and slopes is crucial for solving geometrical problems with algebraic equations.

Coordinate geometry is particularly powerful as it gives clarity on various topics like parallel lines, perpendicularity, and distance between points. Understanding how to use coordinate geometry makes it easier to navigate problems that require finding unknown quantities, just like finding our missing \(y\) given the slope and another point.