Problem 35 - Intermediate Algebra

Question

\(x\) intercept of \(-3\) and slope of \(-\frac{5}{8}\)

Step-by-Step Solution

Answer

The line equation is \(y = -\frac{5}{8}x - \frac{15}{8}\).

Step 1: Understand the Problem

We are given the x-intercept of the line, which is \(-3\), and the slope, which is \(-\frac{5}{8}\). We have to find the equation of the line using these two pieces of information.

Step 2: Use the Point-Slope Form Equation

The point-slope form of the equation of a line is\(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \((x_1, y_1)\) is a point on the line. We have the x-intercept, \((x_1, y_1) = (-3, 0)\), and \(m = -\frac{5}{8}\).

Step 3: Plug Values into the Point-Slope Equation

Substitute the values into the point-slope formula: \[ y - 0 = -\frac{5}{8}(x + 3) \].

Step 4: Simplify the Equation

Simplifying the equation gives: \[ y = -\frac{5}{8}x - \frac{15}{8} \]. This is the slope-intercept form of the line, \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept.

Key Concepts

Point-Slope Form

The point-slope form is a way to write the equation of a line when you know the slope and a single point on the line. It's incredibly useful for converting real-world information directly into a mathematical model. The general formula for the point-slope form is:

\[ y - y_1 = m(x - x_1) \]
where:

• \( m \) is the slope of the line.
• \((x_1, y_1)\) is a point through which the line passes.

You can think of this formula as capturing the essence of a straight line by defining both its tilt (the slope) and a point it must pass through.

To apply this formula, simply plug in the values you know for the slope and for the coordinates of the point. In our example, the coordinates correspond to an x-intercept \((-3, 0)\) and a slope of \(-\frac{5}{8}\). By inserting these values into the formula, you get a specific equation that describes that particular line on a graph.

Slope-Intercept Form

The slope-intercept form is one of the most recognized forms of a linear equation. It clearly shows two crucial characteristics of a line: its slope and y-intercept. The general structure is:

\[ y = mx + b \]
where:

• \( m \) signifies the slope.
• \( b \) represents the y-intercept, the point where the line crosses the y-axis.

In this form, it’s straightforward to identify both how steep the line is and where it begins on the graph. After simplifying the point-slope form equation from our exercise, that's exactly the form we ended up with:

\[ y = -\frac{5}{8}x - \frac{15}{8} \]
Here, \( m = -\frac{5}{8} \), and \( b = -\frac{15}{8} \). The equation translates graphically into a line with a slope of \(-5/8\) that crosses the y-axis at the point \( (0, -\frac{15}{8}) \).
This form is especially helpful when you want to quickly sketch out a line or understand its main attributes at a glance.

Linear Equation

Linear equations are equations of the first degree, which means they involve variables raised only to the first power. In simpler terms, these equations form straight lines when graphed on a coordinate system. They typically look something like:

\[ ax + by = c \]
where:

• \( a \), \( b \), and \( c \) are constants.
• \( x \) and \( y \) are variables.

The uniqueness of a linear equation is its consistency; it has a constant rate of change, indicated by its slope. In our exercise, we derive such an equation by starting with familiar formulas like point-slope and slope-intercept. Each form can be transformed into others, depending on what information is initially available and what results you need.
Understanding linear equations is essential for numerous areas in both math and its applications, from simple calculations in everyday life to complex analysis in science and engineering. They are foundational to the study of algebra, serving as a gateway to more complex mathematical concepts.