Problem 37 - Elementary Algebra

Question

Determine the slope and \(y\) intercept of the line represented by the given equation, and graph the line. (Objective 2) $$-4 x+9 y=18$$

Step-by-Step Solution

Answer

Slope: \(\frac{4}{9}\); Y-intercept: \(2\).

Step 1: Convert into Slope-Intercept Form

The standard form of a line is given by the equation \(Ax + By = C\). First, we need to transform the given equation \(-4x + 9y = 18\) into slope-intercept form, \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. To do this, solve for \(y\) by isolating it on one side of the equation. Start by adding \(4x\) to both sides: \[9y = 4x + 18\]. Then, divide every term by \(9\): \[y = \frac{4}{9}x + 2\].

Step 2: Identify the Slope and Y-intercept

Now that the equation is in slope-intercept form \(y = mx + b\), identify "\(m\)" as the slope and "\(b\)" as the y-intercept. For \(y = \frac{4}{9}x + 2\), the slope \(m\) is \(\frac{4}{9}\) and the y-intercept \(b\) is \(2\).

Step 3: Graph the Line

To graph the line, first plot the y-intercept on the graph. Since the y-intercept is \(2\), this point is \((0, 2)\). Next, use the slope \(\frac{4}{9}\). The slope tells us that for every 9 units moved horizontally (to the right), the line moves 4 units vertically (upwards). From the point \((0, 2)\), move right 9 units and up 4 units to another point \((9, 6)\). Plot this point and draw a line through both points to graph the line.

Key Concepts

Linear Equations

A linear equation is an equation that makes a straight line when it is graphed. These equations typically appear in the form of the standard equation, written as \( Ax + By = C \). Here, \( A \), \( B \), and \( C \) are constants, and \( x \) and \( y \) are variables. The equation describes a direct relationship between \( x \) and \( y \).To better understand how this works, let's look at our example equation: \(-4x + 9y = 18\). This is set in the standard form. The ultimate goal is often to express it in the slope-intercept form, \( y = mx + b \), which is more user-friendly for identifying the slope and intercept of the line.

Graphing Lines

Graphing lines is a way to visualize the solutions of a linear equation. To graph a line, you'll need the equation in the slope-intercept form: \( y = mx + b \). This allows you to quickly gather two key pieces of information:

• **Slope (\( m \))**: indicates how steep the line is.
• **Y-intercept (\( b \))**: the point where the line crosses the y-axis.

For example, in the equation \( y = \frac{4}{9}x + 2 \), the y-intercept is \( 2 \). This means the line crosses the y-axis at the point \((0, 2)\). The slope \( \frac{4}{9} \) means the line rises \( 4 \) units for every \( 9 \) units it runs horizontally. Start graphing by plotting the y-intercept, then use the slope to find another point. Connect these points to reveal the line.

Algebraic Manipulation

Algebraic manipulation is an important skill for rewriting equations in a more useful form. It involves using operations like addition, subtraction, multiplication, and division to rearrange an equation. In the context of linear equations, algebraic manipulation is especially useful for converting from the standard form to the slope-intercept form.Take the equation \(-4x + 9y = 18\). Starting with the standard form, you want to isolate \( y \) on one side:- Add \( 4x \) to both sides to have the \( x \) terms on the right: \( 9y = 4x + 18 \).- Then, divide every term by \( 9 \) to solve for \( y \): \( y = \frac{4}{9}x + 2 \).This process makes it clear what the slope and y-intercept are, assisting in graphing the line. Understanding and applying these manipulations is crucial for solving many algebra problems.