Problem 84 - Precalculus Essentials

Question

If one point on a line is (2,-6) and the line's slope is \(-\frac{3}{2},\) find the \(y\)-intercept.

Step-by-Step Solution

Answer

The y-intercept of the line is -3.

Step 1: Write the slope-intercept form of the equation

The general slope-intercept form of the equation of a line is \(y = mx+b\). In this problem, we are given the slope \(m = -\frac{3}{2}\) and a point \((x,y) = (2,-6)\). Substituting these values into the equation, we get \(-6 = \left(-\frac{3}{2}\right)(2) + b\).

Step 2: Solve for b

After substituting, simplify the equation to solve for \(b\). This simplification gives \(-6 = -3 + b\). Solving for \(b\) requires adding 3 to both sides of the equation, giving \(b = -6 + 3 = -3\).

Key Concepts

Understanding Slope-Intercept Form

The slope-intercept form is a way of writing the equation of a straight line. It follows the format \( y = mx + b \). Here, \( y \) and \( x \) represent the coordinates of any point on the line.
The letter \( m \) stands for the slope of the line, which represents the line's steepness or incline. The slope is basically the "rise over run," meaning how much the line goes up (or down) for each step to the right. 💡 Think of it like climbing steps: how much you go up with each step forward.
The \( b \) value in this equation is the \( y \)-intercept. It's the point where the line crosses the \( y \)-axis. In simple terms, it's where the line meets the vertical axis when \( x = 0 \). This form is widely used because it gives both the slope and the \( y \)-intercept directly from the equation.

Grasping Linear Equations

A linear equation, such as \( y = mx + b \), defines a straight line when plotted on a graph. It's made up of two variables, \( x \) and \( y \), which can vary, and constants \( m \) and \( b \), which remain fixed for any given line.
The beauty of linear equations is their simplicity and consistency. They make predicting values straightforward because the relationship between \( x \) and \( y \) is always linear, meaning without any curves or bends.
These equations are foundational in algebra and are used to solve real-world problems. Whenever you see a relationship that consistently raises or lowers by the same amount, think of linear equations. They're like the algebraic equivalent of a subway line or a railway track—steady and predictable straight lines.

Steps to Solve for b

In this problem, finding the \( y \)-intercept \( b \) involves substituting the known values into the slope-intercept form and solving an equation. Here's the scoop on how to do it:

• We start with the equation \( y = mx + b \).
• You plug in the known slope \( m = -\frac{3}{2} \) and the given point \((x, y) = (2, -6)\).
• You substitute these values into the formula so that \( -6 = (-\frac{3}{2})(2) + b \).
• Now, simplify on the right side to get \(-6 = -3 + b \).
• To find b, solve like any regular equation! Add 3 to both sides to isolate \( b \) on one side, resulting in \( b = -3 \).

That answer gives you the full description of the line's equation: \( y = -\frac{3}{2}x - 3 \). It tells us everything we need to know about that line's position and direction on a graph. Solving for \( b \) is like finding the last piece of the puzzle to draw the line confidently across your graph paper! 😊