X And Y Intercepts: Solve 2x + 3y = 34 Easily

Hey guys! Let's dive into finding the $x$- and $y$-intercepts of the graph represented by the equation $2x + 3y = 34$. We'll break it down step by step so it's super easy to follow. Remember, the $x$- and $y$-intercepts are simply the points where the line crosses the $x$-axis and $y$-axis, respectively. Knowing how to find these intercepts is fundamental in algebra and helps in graphing linear equations quickly.

Understanding Intercepts

Before we get started, let's make sure we all understand what intercepts are. The x-intercept is the point where the line crosses the x-axis. At this point, the y-coordinate is always zero. Similarly, the y-intercept is where the line crosses the y-axis, and at this point, the x-coordinate is always zero. Understanding this basic concept is crucial for solving the problem at hand.

The Significance of Intercepts

Intercepts provide key insights into the behavior of a linear equation. They tell us where the line starts and ends relative to the axes, which can be extremely useful in real-world applications. For example, if this equation represented a budget constraint, the intercepts would tell us the maximum amount of each item we could purchase if we spent all our money on just that item. This kind of interpretation makes understanding intercepts not just an academic exercise, but a practical skill.

Finding the $x$-intercept

To find the $x$-intercept, we need to set $y = 0$ in the equation and solve for $x$. So, let's do that:

$2x + 3y = 34$

Substitute $y = 0$:

$2x + 3(0) = 34$

Simplify:

$2x = 34$

Now, divide both sides by 2:

$x = \frac{34}{2}$

$x = 17$

So, the $x$-intercept is 17. This means the line crosses the $x$-axis at the point $(17, 0)$.

Detailed Steps for Finding the X-intercept

Let's break down each step to ensure clarity.

1. Set y to Zero:

   We begin by setting $y = 0$. This is because, at any point on the x-axis, the y-coordinate is always zero. By substituting $y = 0$ into the equation, we effectively restrict our focus to the x-axis.

2. Substitute and Simplify:

   Substituting $y = 0$ into the equation $2x + 3y = 34$ gives us $2x + 3(0) = 34$. Simplifying this, we get $2x = 34$, as $3(0)$ is simply zero.

3. Isolate x:

   To find the value of $x$, we need to isolate it on one side of the equation. We do this by dividing both sides of the equation $2x = 34$ by 2. This gives us $x = \frac{34}{2}$.

4. Solve for x:

   Finally, we solve for $x$ by performing the division. $\frac{34}{2}$ simplifies to 17. Therefore, $x = 17$.

Thus, the x-intercept is 17, indicating that the line intersects the x-axis at the point (17, 0).

Finding the $y$-intercept

To find the $y$-intercept, we need to set $x = 0$ in the equation and solve for $y$. Here we go:

$2x + 3y = 34$

Substitute $x = 0$:

$2(0) + 3y = 34$

Simplify:

$3y = 34$

Now, divide both sides by 3:

$y = \frac{34}{3}$

So, the $y$-intercept is $\frac{34}{3}$. This means the line crosses the $y$-axis at the point $(0, \frac{34}{3})$.

Step-by-Step Guide to Finding the Y-intercept

1. Set x to Zero:

   To find the y-intercept, we set $x = 0$. This is because any point on the y-axis has an x-coordinate of zero. By doing this, we narrow our focus to the y-axis.

2. Substitute and Simplify:

   Substituting $x = 0$ into the equation $2x + 3y = 34$ yields $2(0) + 3y = 34$. Simplifying this, we get $3y = 34$, since $2(0)$ equals zero.

3. Isolate y:

   To find the value of $y$, we need to isolate it. We achieve this by dividing both sides of the equation $3y = 34$ by 3. This results in $y = \frac{34}{3}$.

4. Solve for y:

   Finally, we solve for $y$ by performing the division. The fraction $\frac{34}{3}$ is already in its simplest form, as 34 and 3 have no common factors other than 1. Therefore, the y-intercept is $\frac{34}{3}$.

Thus, the y-intercept is $\frac{34}{3}$, meaning the line intersects the y-axis at the point $(0, \frac{34}{3})$.

State the Answers

Alright, let's state our answers clearly:

• The $x$-intercept is 17.
• The $y$-intercept is $\frac{34}{3}$.

These are the points where the line $2x + 3y = 34$ intersects the $x$-axis and $y$-axis, respectively. Great job!

Why These Answers Matter

Understanding where a line crosses the x and y axes is incredibly useful. For example, in economics, these intercepts can represent the maximum quantities of goods a consumer can buy with a fixed budget. In physics, they might indicate initial conditions or endpoints of a process. The ability to quickly find and interpret these intercepts is a valuable skill in many fields.

Graphing the Line

Knowing the intercepts makes graphing the line super easy. Just plot the points $(17, 0)$ and $(0, \frac{34}{3})$ on a graph and draw a line through them. You've got your graph! These two points are sufficient to define the line uniquely.

Step-by-Step Graphing Using Intercepts

1. Plot the X-Intercept:

   Locate the point on the x-axis where $x = 17$. Mark this point, which is (17, 0).

2. Plot the Y-Intercept:

   Find the point on the y-axis where $y = \frac{34}{3}$, which is approximately 11.33. Mark this point, which is $(0, \frac{34}{3})$.

3. Draw the Line:

   Use a ruler or straight edge to draw a straight line that passes through both the x-intercept (17, 0) and the y-intercept $(0, \frac{34}{3})$.

That's it! You've successfully graphed the line using its intercepts. This method is straightforward and efficient for linear equations.

Conclusion

So, to wrap things up, finding the $x$- and $y$-intercepts involves setting $y = 0$ to find the $x$-intercept and setting $x = 0$ to find the $y$-intercept. We found that the $x$-intercept is 17 and the $y$-intercept is $\frac{34}{3}$. These intercepts are crucial for graphing the line and understanding its behavior.

Final Thoughts on Mastering Intercepts

Mastering the concept of intercepts is a cornerstone of algebra. It provides a simple yet powerful method for understanding and visualizing linear equations. Keep practicing, and you'll find that this skill becomes second nature. Remember, the x-intercept is where y equals zero, and the y-intercept is where x equals zero. This simple rule will guide you through countless problems. Keep up the great work, guys!