Graph $3x - 4y = 24$.

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01-11-2026
Mathematics

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Graph $3x - 4y = 24$.

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-01T13:49:41-05:00

Identify the x-intercept by setting $y=0$, yielding $x=8$.
Identify the y-intercept by setting $x=0$, yielding $y=-6$.
Express the equation in slope-intercept form as $y = \frac{3}{4}x - 6$ to find the slope of $\frac{3}{4}$.
Plot the points $(8, 0)$ and $(0, -6)$ and draw a line through them to represent the linear equation $\boxed{3x - 4y = 24}$.

Explanation

Analyzing the Linear Equation

The given equation is $3x - 4y = 24$, which is a linear equation in two variables representing a straight line in the Cartesian coordinate system. To graph this line, we need to identify key points that lie on it, specifically the x-intercept and the y-intercept. These intercepts provide the points where the line crosses the axes, making it easy to plot.

Finding the Intercepts

To find the x-intercept, we set $y = 0$ and solve for $x$:
\[3x - 4(0) = 24\]
\[3x = 24\]
\[x = 8\]
So, the x-intercept is at the point $(8, 0)$.

To find the y-intercept, we set $x = 0$ and solve for $y$:
\[3(0) - 4y = 24\]
\[-4y = 24\]
\[y = -6\]
So, the y-intercept is at the point $(0, -6)$.

Determining the Slope-Intercept Form

We can also express the line in slope-intercept form ($y = mx + b$) to understand its slope and orientation.
Starting from $3x - 4y = 24$:
\[-4y = -3x + 24\]
Dividing by -4:
\[y = \frac{-3}{-4}x + \frac{24}{-4}\]
\[y = \frac{3}{4}x - 6\]
This confirms the y-intercept is $-6$ and the slope of the line is $\frac{3}{4}$. This means that for every 4 units we move to the right, the line moves 3 units upwards.

Graphing the Line

To graph the equation, we plot the two points $(8, 0)$ and $(0, -6)$ on the Cartesian plane. By connecting these points with a straight line, we obtain the graphical representation of the equation $3x - 4y = 24$. This line extends indefinitely in both directions. The final result of the graphing process is a straight line passing through $(0, -6)$ and $(8, 0)$.

Examples

In architecture and engineering, drawing linear relationships is crucial. For instance, if a construction budget requires a specific ratio of materials (like concrete versus steel) that costs exactly a set amount, it can be modeled as a linear equation. Graphing this equation allows builders to visualize all possible combinations of materials that fit within that budget, helping them choose the most cost-effective and structurally sound proportions.