Problem 32 - Intermediate Algebra

Question

Find \(y\) if the line through \((5,2)\) and \((-3, y)\) has a slope of \(-\frac{7}{8}\)

Step-by-Step Solution

Answer

The value of \(y\) is -5.

Step 1: Understand the Slope Formula

The slope of a line can be calculated using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\), where \((x_1, y_1)\) and \((x_2, y_2)\) are two points on the line, and \(m\) is the slope.

Step 2: Substitute Known Values

Substitute the given points \((5, 2)\) for \((x_1, y_1)\) and \((-3, y)\) for \((x_2, y_2)\) into the slope formula, along with the given slope \(m = -\frac{7}{8}\). This sets up the equation: \(-\frac{7}{8} = \frac{y - 2}{-3 - 5}\).

Step 3: Simplify the Denominator

Calculate \(-3 - 5\) to get \(-8\). So the equation becomes \(-\frac{7}{8} = \frac{y - 2}{-8}\).

Step 4: Solve for \(y\)

Since both sides of the equation have \(-8\) in the denominator, you can set the numerators equal to each other: \(-7 = y - 2\).

Step 5: Isolate \(y\)

Add 2 to both sides of the equation:\(-7 + 2 = y\),resulting in \(y = -5\).

Key Concepts

Slope Formula in Algebra

The slope formula is a central concept in algebra that helps us determine how steep a line is, or its inclination. The slope, often represented by the letter \( m \), provides a constant ratio showing the change in \( y \) for every change in \( x \). The formula is expressed as:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]Here, \((x_1, y_1)\) and \((x_2, y_2)\) are two distinct points on a line. This formula essentially finds the 'rise' over the 'run,' or the vertical change over the horizontal change.

• Positive Slope: The line rises from left to right.
• Negative Slope: The line falls from left to right.
• Zero Slope: The line is perfectly horizontal.
• Undefined Slope: The line is perfectly vertical.

Using the slope formula is straightforward. It gives us a way to quantify the angle or steepness of any straight line simply by using points from a line. This makes it an invaluable tool in coordinate geometry.

Understanding Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This branch of mathematics combines algebra and geometry to explore points, lines, and surfaces.In coordinate geometry, every point on a plane is identified using an ordered pair of numbers, known as coordinates. Each point is expressed as \((x, y)\), where \(x\) is the horizontal position and \(y\) is the vertical position. Some key elements of coordinate geometry include:

• Points: Defined by coordinates \((x, y)\).
• Lines: Can be represented in multiple forms, such as slope-intercept \( y = mx + c \) or standard form \( Ax + By = C \).
• Distance: The length between two points, calculated using the distance formula.
• Midpoint: The middle point of a line segment, calculated using the midpoint formula.

Coordinate geometry is powerful because it allows for the geometric visualization of algebraic equations, providing a deep understanding of spatial relationships.

Problem Solving Steps in Coordinate Geometry

Solving algebra problems in coordinate geometry often involves a series of logical steps to reach the solution. These steps ensure you accurately find variables such as slope or coordinates in a line, making problem-solving systematic and clear.One effective strategy is to first identify the given information in the problem. For example, in the provided exercise, knowing the points \((5,2)\) and \((-3, y)\), along with the slope \(-\frac{7}{8}\), sets a foundation.To solve such problems:

• Use the Slope Formula: Begin by substituting the known values into the slope formula.
• Simplify Expressions: Perform any necessary arithmetic operations to simplify the equation.
• Isolate Variables: Solve for the unknown by isolating it on one side of the equation.
• Verify Solutions: Always check your results by substituting them back into the original equation to ensure accuracy.

This step-by-step approach not only helps in finding the right answer but also builds critical thinking skills that are essential for tackling more complex equations in the future.