Problem 91 - Algebra 1: Concepts and Skills

Question

Write the equation in slope-intercept form. Then graph the equation. (Lesson 4.7) $$4 x+2 y-12=0$$

Step-by-Step Solution

Answer

The equation in slope-intercept form is \(y = -2x + 6\). The slope is -2 and the y-intercept is 6. The graph of the equation would start at the point (0,6) on the y-axis and fall to the right.

Step 1: Rearrange the equation

Rearrange the equation by isolating the \(y\) term on one side. That will involve moving \(4x\) from the left side to the right side by subtracting \(4x\) from both sides and then dividing through by 2. The equation becomes: \(2y = -4x+12\), or simplified to \(y = -2x+6\).

Step 2: Identify the slope and y-intercept

Now that the formula is in slope-intercept form, you can easily identify the slope and the y-intercept. The slope, \(m\), is the coefficient of \(x\), which is -2, and the y-intercept, \(b\), is the constant term, which is 6.

Step 3: Graph the equation

To graph this equation, first locate the y-intercept on the y-axis. This is the point (0,6). Then use the slope to determine the direction of the line. Since the slope is negative, the line will fall as you move to the right. From the y-intercept, move down 2 units and to the right 1 unit to plot the next point. You can continue this pattern to draw the line.

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form of a linear equation is like holding a key to graphing lines quickly. It's written as y = mx + b, where m represents the slope of the line, and b indicates the y-intercept.

Let's dive into an example. Starting from the given equation 4x + 2y - 12 = 0, our goal is to reformat it into the slope-intercept form. By isolating y, we get y = -2x + 6. Here, the coefficient -2 is the slope, and the constant 6 is our y-intercept. This form tells us at a glance how the line will look when graphed—which is on a steady decline as it moves from left to right, due to the negative slope.

Slope-intercept form simplifies graphing because it gives us critical information about the line without having to calculate additional points.

Identifying Slope and Y-Intercept

Moving on to identifying the slope and y-intercept from the slope-intercept form, let's recall our equation y = -2x + 6. Here, the slope m equals -2. The slope dictates the steepness and direction of the line on the graph. A negative slope means the line decreases from left to right, and the larger the absolute value of the slope, the steeper the line.

The y-intercept is the point at which the line crosses the y-axis, and for our equation, it is given by b as 6. This means our line will cross the y-axis at (0,6). Always remember that the y-intercept is an actual point on the graph and is invaluable for beginning to plot your line. With the slope and y-intercept in hand, graphing becomes a straightforward task that follows predictable rules and creates a visual representation of the equation.

Graphing Techniques in Algebra

Let's talk about the third step in our problem-solving journey—graphing the equation using algebraic techniques. Once the y-intercept (0,6) is placed on the graph, we use the slope to find our next point. The slope of -2 is interpreted as rise over run, meaning we go down 2 units (since it's negative) and move 1 unit to the right.

Repeat the Pattern

After plotting the second point, we extend this pattern to find more points, ensuring our line is accurate. Repeat these steps to create as many points as needed to form a straight line.

Draw the Line

Next, we connect the dots. A ruler is a handy tool to maintain the precision needed for a straight line. This visual display is the culmination of our algebraic expression—it shows all points that satisfy the equation y = -2x + 6.

Check Your Work

Always verify a couple of points on your line by plugging their x and y values back into the original equation. If the equation holds true, you’ve graphed it correctly. Remember, graphing is not just a mechanical process but also a check for understanding the relationship between algebraic equations and their graphical counterparts.