How can I determine which graph represents the equation 12 = 3x + 4y?

You can rewrite the equation in slope-intercept form (y = mx + b) and compare it to the graphs. For 12 = 3x + 4y, solving for y gives y = (12 - 3x) / 4. Then, check which graph matches this line by verifying points or slope and intercept.

What is the slope and y-intercept of the line 12 = 3x + 4y?

Rearranging the equation to slope-intercept form yields y = - (3/4)x + 3. Therefore, the slope is -3/4 and the y-intercept is at (0, 3). The correct graph should reflect these features.

How can plotting points help identify the correct graph for 12 = 3x + 4y?

By choosing specific x-values, calculating corresponding y-values, and plotting these points, you can see which graph passes through these points. The graph that contains points such as (0, 3), (4, 0), and (-4, 6) matches the equation.

Why is it important to check the intercepts when identifying the graph of 12 = 3x + 4y?

Because the intercepts directly relate to the equation's structure, verifying the x-intercept (when y=0) and y-intercept (when x=0) helps confirm which graph accurately represents the line. For this equation, x-intercept is at (4, 0), and y-intercept at (0, 3).

Can the equation 12 = 3x + 4y be represented by a vertical or horizontal line?

No, because the equation is linear with both x and y variables, resulting in a sloped line. Vertical lines have equations of the form x = constant, and horizontal lines have y = constant; this equation does not fit either form.

Which Graph Represents The Equation 12 = 3x + 4y? Explain How You Know.

Which Graph Represents The Equation 12 = 3x + 4y? Explain How You Know.

Understanding how to identify the correct graph that corresponds to a given linear equation is a fundamental skill in algebra and coordinate geometry. When presented with the equation 12 = 3x + 4y, students and learners often wonder how to determine which graph visually represents this equation. This article will guide you through the process of analyzing the equation, converting it into a slope-intercept form, and examining the key features that help you recognize the right graph. By the end, you'll understand not only which graph matches the equation but also why it does, based on the equation's properties.

Understanding the Equation 12 = 3x + 4y

Before selecting the correct graph, it is crucial to understand the structure of the equation itself. The equation 12 = 3x + 4y is linear, meaning it graphs as a straight line on the coordinate plane. To analyze it effectively, we should rewrite it in a more familiar form, such as the slope-intercept form y = mx + b.

Converting the Equation to Slope-Intercept Form

The current form of the equation is:

12 = 3x + 4y

To express it as y = mx + b, follow these steps:

Isolate the term with y: Subtract 3x from both sides:

12 - 3x = 4y

Divide both sides by 4 to solve for y:

y = (12 - 3x) / 4

Rewrite the equation to separate the terms:

y = (12 / 4) - (3x / 4)

Simplify:

y = 3 - (3/4)x

Now, the equation is in slope-intercept form:

y = - (3/4)x + 3

Key features of this form:

Slope (m): -3/4

Y-intercept (b): 3

How To Recognize the Correct Graph

Having the slope and y-intercept allows us to identify the correct graph visually. Here are the main features to look for:

1. Y-Intercept

The y-intercept is the point where the line crosses the y-axis.
For this equation, the y-intercept is at (0, 3).

When examining graphs, find the line that crosses the y-axis at y = 3.

2. Slope

The slope indicates the steepness and direction of the line.
A slope of -3/4 means:
The line decreases as x increases.
For every increase of 4 units in x, y decreases by 3 units.
The negative sign indicates a downward slope from left to right.

On the graph, check for a line that goes downward from left to right with a gentle slope.

3. Additional Points

To confirm, identify additional points on the line.
For example, plug in x = 4:

y = - (3/4)(4) + 3 = -3 + 3 = 0

So, the point (4, 0) should lie on the line.
Similarly, for x = 0:

y = 3 (the y-intercept point).

For x = -4:

y = - (3/4)(-4) + 3 = 3 + 3 = 6

The point (-4, 6) should also be on the line.

Use these points to verify the line's position on the graph.

Steps to Identify the Correct Graph

To determine which graph matches the equation, follow these steps:

Step 1: Find the Y-Intercept

Locate the point where the line crosses the y-axis.
Confirm whether it is at (0, 3).

Step 2: Check the Slope

From the y-intercept, count the rise and run to see if it matches the slope of -3/4.
For example, move 4 units to the right (positive x direction), and see if the line goes down 3 units.

Step 3: Verify Additional Points

Calculate points like (4, 0) and (-4, 6).
Ensure these points lie on the line in the graph.

Step 4: Confirm the Line's Direction and Steepness

The line should slope downward from left to right.
Its steepness should correspond to a slope of -3/4.

Common Mistakes and How to Avoid Them

While analyzing graphs, a few common pitfalls can lead to incorrect conclusions:

1. Confusing the Slope Sign

Make sure to note whether the slope is positive or negative.
A positive slope slopes upward, negative slopes slope downward.

2. Misreading the Y-Intercept

Confirm the point where the line crosses the y-axis.
Sometimes, multiple lines may cross close to y=3, so check carefully.

3. Overlooking Additional Points

Use multiple points to verify the line's position.
Relying solely on the y-intercept can be misleading if the graph is misdrawn.

Conclusion: Which Graph Represents 12 = 3x + 4y?

Based on the analysis above, the graph that correctly represents the equation 12 = 3x + 4y is the one that:

Crosses the y-axis at (0, 3)

Has a slope of -3/4, descending gently from left to right

Passes through points like (4, 0) and (-4, 6)

By applying the process of converting the equation into slope-intercept form, identifying key features, and verifying multiple points, you can confidently select the correct graph. Remember, understanding the fundamental properties of linear equations—such as slope and intercepts—is essential for interpreting and visualizing algebraic relationships accurately.

Whether you're studying for exams, solving real-world problems, or just enhancing your math skills, mastering how to connect equations to their graphs is a valuable tool. Keep practicing with different equations, and soon you'll find it easier to recognize their graphical representations with confidence.