Problem 31 - Algebra for College Students

Question

Find \(x\) if the line through \((x, 4)\) and \((2,-5)\) has a slope of \(-\frac{9}{4}\)

Step-by-Step Solution

Answer

The value of \(x\) is -2.

Step 1: Recall the Formula for Slope

The slope formula for a line passing through two points, \((x_1, y_1)\) and \((x_2, y_2)\), is given by \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Here, the slope \(m\) is known to be \(-\frac{9}{4}\).

Step 2: Substitute Values into the Slope Formula

Identify the coordinates: \((x_1, y_1) = (x, 4)\) and \((x_2, y_2) = (2, -5)\). Substitute these into the slope formula to get:\[\frac{-5 - 4}{2 - x} = -\frac{9}{4}.\]

Step 3: Simplify the Expression

Calculate \(-5 - 4\) to get \(-9\). So our equation now becomes:\[\frac{-9}{2-x} = -\frac{9}{4}.\]

Step 4: Solve for \(x\)

Since the numerators are the same, set the denominators equal:\[2 - x = 4.\]Now solve for \(x\) by rearranging the equation:\[2 = 4 + x.\]Subtract 4 from both sides:\[2 - 4 = x\]So, \[x = -2.\]

Step 5: Conclusion

The value of \(x\) is \(-2\), which satisfies the given slope condition.

Key Concepts

Slope Formula

Understanding the slope formula is essential when working with linear equations. The slope, often denoted as \(m\), represents the steepness or incline of a line between two points. The formula is given by:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]This represents the ratio of the vertical change (also known as rise) to the horizontal change (known as run) between two points \((x_1, y_1)\) and \((x_2, y_2)\).

• The numerator, \(y_2 - y_1\), captures the change in the \(y\)-coordinates between the two points, and
• the denominator, \(x_2 - x_1\), captures the change in the \(x\)-coordinates.

The slope formula is crucial because it allows you to analyze and interpret the behavior of a line based solely on its endpoints. A positive slope means the line ascends from left to right, while a negative slope, like \(-\frac{9}{4}\) in our exercise, indicates that the line descends from left to right.

Coordinate Geometry

Coordinate Geometry, or analytic geometry, lets us study geometric figures using an ordered pair system. This method merges algebra and geometry, enabling us to solve problems involving points, lines, and shapes in a plane. When it comes to linear equations, coordinate geometry gives us the framework to visualize and compute distances and slopes through the use of coordinate pairs.
In our exercise, we used the coordinates \((x, 4)\) and \((2, -5)\). Each coordinate pair consists of an \(x\) and a \(y\) value, which designates their location on the Cartesian plane. This allows for practical application of algebraic formulas, like the slope formula, to determine key characteristics about the line they form.

• This visualization helps us better understand how two points can determine a line entirely.
• We can predict line behavior, analyze intersection points, and graph these lines to relate them back to real-world problems.

Problem Solving Steps

Following a structured approach can significantly aid in solving problems. Here are the steps we used:
1. Identify the Formula: Start with understanding what is needed, in this case, the slope formula which connects slope to two points on a line.
2. Substitute Known Values: Plug in the points you know into the formula. In our case, \((x, 4)\) and \((2, -5)\), along with the slope \(-\frac{9}{4}\).
3. Simplify and Rearrange: Perform basic algebra to simplify and adjust the equation to isolate the desired variable. Here, we simplified the equation to \(\frac{-9}{2-x} = -\frac{9}{4}\).
4. Solve for the Unknown: manipulate the equation to find the unknown by equating both sides properly. This helped us find \(x = -2\).
5. Confirm Solution: Double-check by plugging values back into the original equation to ensure the solution fits the conditions given, ensuring accuracy and validity in problem-solving.

• Practice these steps consistently to enhance problem-solving skills.
• Never rush in solving; verify each step to confirm correctness.