Problem 71 - Algebra for College Students

Question

Find \(k\) so that the line through \((2, k)\) and \((-3,-5)\) has slope \(\frac{1}{2}\)

Step-by-Step Solution

Answer

The value of \( k \) is \( -\frac{5}{2} \).

Step 1: Understand the slope formula

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

Step 2: Plug in given points and slope

Substitute the given points \( (2, k) \) and \((-3, -5)\) into the slope formula, along with the given slope \( \frac{1}{2} \). \(\frac{-5 - k}{-3 - 2} = \frac{1}{2} \).

Step 3: Simplify the formula

Simplify the denominator: \( -3 - 2 = -5 \). This gives the equation: \(\frac{-5 - k}{-5} = \frac{1}{2} \).

Step 4: Solve for k

To clear the fraction, multiply both sides by \( -5 \): \((-5 - k) = \frac{1}{2} \times -5\) Simplify: \(-5 - k = -\frac{5}{2} \). Next, add 5 to both sides: \(- k = -\frac{5}{2} + 5 \). Finally, simplify to find \( k \): \(- k = -\frac{5}{2} + \frac{10}{2}\ = \frac{5}{2}\) Divide by -1 to solve for \( k \): \( k = -\frac{5}{2} \).

Key Concepts

slope formula

The slope formula is a way to measure the steepness and direction of a line. It tells us how much the y-coordinate (or vertical position) of a point on the line changes for every single unit of change in the x-coordinate (or horizontal position). This is why it's represented as 'rise over run'. You can calculate the slope between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \].
Here:

• \(m\) is the slope
• \(x_1\), \(y_1\) are the coordinates of the first point
• \(x_2\), \(y_2\) are the coordinates of the second point

The slope tells us a lot about the line:

• If the slope is positive, the line goes upwards as you move to the right.
• If the slope is negative, the line goes downwards as you move to the right.
• If the slope is zero, the line is horizontal.
• If the slope is undefined (division by zero), the line is vertical.

Knowing the slope of a line is very important in coordinate geometry and in solving linear equations.

linear equations

Linear equations are equations of the first degree, meaning they have no exponents greater than 1. A typical linear equation looks like this: \[ y = mx + c \] Here:

• \( y \) is the dependent variable.
• \( x \) is the independent variable.
• \( m \) is the slope.
• \( c \) is the y-intercept, the point where the line crosses the y-axis.

In solving linear equations, we find the relationship between the variables. For example, if we know the coordinates of two points on a line, we can determine the slope using the slope formula. Then, we can use the slope to form the equation of the line. Linear equations are easy to solve and form the basis of more complex mathematical concepts. They are used in various real-world applications like calculating speed, determining cost, and measuring distances.

coordinate geometry

Coordinate geometry, also called analytic geometry, is a branch of mathematics that uses algebra to study geometric problems. It involves the coordinate plane which is formed by two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). Every point on this plane is defined by a pair of coordinates \( (x, y) \). Key concepts in coordinate geometry include:

• The distance formula: used to calculate the distance between two points.
• The midpoint formula: used to find the midpoint of a segment connecting two points.
• The slope formula: as explained earlier, used to measure the steepness of a line connecting two points.

Coordinate geometry forms the basis of calculus and other branches of mathematics. It allows us to analyze and solve geometrical problems using algebraic methods. This makes it a powerful tool in mathematics, engineering, physics, and many other fields.