Problem 72 - Algebra 1: Concepts and Skills

Question

Find the slope and the y-intercept of the graph of the equation. Then graph the equation. $$ 4 y+12 x=16 $$

Step-by-Step Solution

Answer

The slope of the line is -3 and the y-intercept is 4. The graph would start at the point (0,4) and slope downwards from left to right.

Step 1: Rearrange the Equation

First, rewrite the given equation \(4 y+12 x=16\) to the 'y=mx+b' form. Subtract \(12x\) from both sides of the equation so that only \(4y\) is on the left side of the equation. This gives you the equation \(4 y = -12 x + 16\). Then, divide everything by 4 to isolate \(y\). This results in the equation \( y = -3 x + 4 \).

Step 2: Identify the Slope and Y-Intercept

Now that the equation is in that form, the coefficients are easily identifiable. The coefficient of \(x\) (-3) in the equation \( y = -3 x + 4 \) is the slope and the constant term (4) is the y-intercept.

Step 3: Graph the Equation

To graph the equation \( y = -3 x+ 4 \), start at the y-intercept (0,4) on the graph. Then, use the slope to determine the next point. Since the slope is negative, move down 3 units (the denominator of the slope) and to the right 1 unit (the numerator of the slope). Repeat this process to draw the line.

Key Concepts

Rearranging Equations

Rearranging equations is the art of transforming a given equation into a more useful or understandable form. One of the most common forms for linear equations is the slope-intercept form, represented as \( y = mx + b \). Here, \( m \) is the slope and \( b \) is the y-intercept. To achieve this form, you typically need to isolate \( y \) on one side of the equation.

• Start by moving any other terms to the opposite side, often done by addition or subtraction.
• Once these terms have been relocated, divide by the coefficient attached to \( y \) to solve for \( y \).

In the given equation \( 4y + 12x = 16 \), we first subtract \( 12x \) from both sides to rearrange it into \( 4y = -12x + 16 \). Then, divide every term by 4 to isolate \( y \), resulting in \( y = -3x + 4 \). This method leaves you with a clean, easy-to-interpret equation to identify both the slope and the y-intercept.

Graphing Linear Equations

Graphing linear equations is a way to visualize mathematical relationships on a coordinate plane. Once you have your equation in the \( y = mx + b \) form, the graphing process becomes more straightforward.

• Begin at the y-intercept, \( b \). This is your starting point on the y-axis.
• From this point, use the slope, \( m \), which is expressed as a fraction \( \frac{rise}{run} \).
• If \( m = ?3 \), rewrite it as \( \frac{-3}{1} \). Here, "rise" is \(-3\) and "run" is \(1\). Move down 3 units on the y-axis and 1 unit to the right along the x-axis to place your next point.

Repeat these steps to find additional points. Draw a line through the points, and extend it across the graph to complete the line representation of the equation.

Identifying Slope and Y-Intercept

Once an equation is in the slope-intercept form \( y = mx + b \), identifying the slope \( m \) and the y-intercept \( b \) is straightforward.

• The coefficient of \( x \) in the equation gives you the slope \( m \). In this case, \( y = -3x + 4 \) has a slope of \(-3\).
• The constant term in the equation represents the y-intercept \( b \). For our equation, the y-intercept is \( 4 \).

The slope tells you the steepness and direction of the line: a negative slope like \(-3\) means the line falls as you move from left to right. The y-intercept indicates where the line crosses the y-axis. Together, these two components define the entire linear graph.