Problem 53 - Algebra and Trigonometry

Question

Find the slope and y-intercept of the line, and draw its graph. $$ 3 x-4 y=12 $$

Step-by-Step Solution

Answer

Slope: \( \frac{3}{4} \), Y-intercept: \( -3 \). Graph the line using these values.

Step 1: Rearrange the Equation into Slope-Intercept Form

The slope-intercept form of a line is given by \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. We start by rearranging the given equation \( 3x - 4y = 12 \) to this form. Subtract \( 3x \) from both sides to get \( -4y = -3x + 12 \). Then, divide every term by \(-4\) to solve for \( y \): \( y = \frac{3}{4}x - 3 \).

Step 2: Identify the Slope and Y-Intercept

From the equation \( y = \frac{3}{4}x - 3 \), we can directly read off the slope and y-intercept. The slope \( m \) is \( \frac{3}{4} \) and the y-intercept \( b \) is \( -3 \).

Step 3: Plot the Y-Intercept on the Graph

To draw the graph of the line, first plot the y-intercept. The point where the line crosses the y-axis is \( (0, -3) \). Place a point on the coordinate plane at this location.

Step 4: Use the Slope to Plot Another Point

The slope of \( \frac{3}{4} \) means that for every 4 units you move to the right (along the x-axis), you move up 3 units (along the y-axis). Starting from \( (0, -3) \), move 4 units right and 3 units up to plot the second point at \( (4, 0) \).

Step 5: Draw the Line

Connect the two points \( (0, -3) \) and \( (4, 0) \) with a straight line. Extend the line in both directions to cover the graph.

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form is essential in algebra, especially when dealing with linear equations. The slope-intercept form of a line is expressed as \( y = mx + b \). This is a standard way of writing the equation of a line so that both the slope and the y-intercept are immediately apparent.

• \( m \) represents the slope. Think of this as the steepness or tilt of the line.
• \( b \) signifies the y-intercept, the point where the line crosses the y-axis.

To convert an equation into this form, isolate the \( y \) variable on one side. This clarity allows you to graph the line quickly by identifying both the steepness of the line and where it begins on the y-axis.

Linear Equations

Linear equations appear as straight lines when graphed. They are algebraic expressions that represent lines, with each term being either a constant or a product of a constant and a variable. Standard forms of linear equations include the slope-intercept form, point-slope form, and standard form.

• The main feature of a linear equation is its constant rate of change, expressed by the slope.
• The equation itself can be manipulated to express this relationship explicitly, often making calculations and graphing easier.

In linear equations, the solution is any ordered pair \((x, y)\) that lies on the line. Understanding how to rearrange these equations into a more usable form, like slope-intercept form, is a valuable skill in algebra.

Graphing Lines

Graphing lines involves using equations to draw a straight path through the coordinate plane. One of the first steps is identifying points that the line passes through. This often begins with determining the y-intercept and using the slope to find additional points.

• Start by plotting the y-intercept, an easy and direct point found in the slope-intercept equation.
• Utilize the slope to calculate another point by moving up or down and then left or right from the y-intercept.
• Draw a straight line through these points to visualize the equation.

By following these steps, you can transform algebraic expressions into understandable visual lines. Graphing helps in identifying the relationship between variables and can make interpreting algebraic results much clearer.