Finding X And Y Intercepts For 2x - 3y = 9 A Step-by-Step Guide
In mathematics, identifying the intercepts of a linear equation is a fundamental skill. Intercepts are the points where a line crosses the x-axis and the y-axis. Understanding how to find these points provides valuable insights into the behavior and graph of the line. In this comprehensive guide, we will walk through the process of finding the x- and y-intercepts of the equation 2x - 3y = 9. By the end of this article, you'll have a clear understanding of the methods involved and be able to apply them to other linear equations.
Understanding Intercepts
Before diving into the solution, it's essential to understand what intercepts are.
- The x-intercept is the point where the line intersects the x-axis. At this point, the y-coordinate is always 0.
- The y-intercept is the point where the line intersects the y-axis. At this point, the x-coordinate is always 0.
Graphically, intercepts give us two crucial points that help in plotting the line on a coordinate plane. Algebraically, they provide key values that satisfy the equation.
The Significance of Intercepts
Intercepts are not just mere points on a graph; they hold significant meaning in various contexts. For instance, in real-world scenarios:
- In economics, the y-intercept of a cost function might represent the fixed costs, while the x-intercept could indicate the break-even point.
- In physics, if the equation represents motion, the intercepts could signify the initial position or the time it takes to reach a certain point.
Understanding intercepts provides a practical way to interpret linear relationships in real-world applications.
Step-by-Step Solution for 2x - 3y = 9
Let's solve for the x- and y-intercepts of the equation 2x - 3y = 9. We'll break this down into two main parts:
- Finding the x-intercept
- Finding the y-intercept
1. Finding the x-intercept
To find the x-intercept, we set y = 0 in the equation and solve for x. This is because, at any point on the x-axis, the y-coordinate is always 0. Here’s the process:
Given equation: 2x - 3y = 9
Step 1: Substitute y = 0
2x - 3(0) = 9
Step 2: Simplify the equation
2x - 0 = 9
2x = 9
Step 3: Solve for x
To isolate x, we divide both sides of the equation by 2:
x = 9 / 2
x = 4.5
Therefore, the x-intercept is the point where the line crosses the x-axis, and its coordinates are (4.5, 0).
Detailed Explanation of Each Step
Step 1: Substitute y = 0
The initial step in finding the x-intercept is to set the y-value to zero. This is a fundamental concept because the x-intercept is, by definition, the point at which the line crosses the x-axis. On the x-axis, every point has a y-coordinate of zero. By substituting y with 0, we are essentially finding the x-value that corresponds to this specific condition.
In the given equation, 2x - 3y = 9, we replace y with 0:
2x - 3(0) = 9
This substitution simplifies the equation and allows us to isolate and solve for x.
Step 2: Simplify the Equation
After substituting y = 0, the equation becomes:
2x - 3(0) = 9
The next step is to simplify the equation by performing the multiplication:
2x - 0 = 9
Since subtracting 0 from any value does not change the value, the equation further simplifies to:
2x = 9
This simplification is crucial as it brings us closer to isolating x and finding its value. By removing the term involving y, we’ve set up a straightforward equation that can be easily solved.
Step 3: Solve for x
The final step in finding the x-intercept is to solve for x. The simplified equation is:
2x = 9
To isolate x, we need to undo the multiplication by 2. We accomplish this by dividing both sides of the equation by 2. This maintains the balance of the equation while isolating the variable we are interested in:
2x / 2 = 9 / 2
This simplifies to:
x = 4.5
Thus, the x-coordinate of the x-intercept is 4.5. Since the y-coordinate is 0 at the x-intercept, the coordinates of the x-intercept are (4.5, 0).
2. Finding the y-intercept
To find the y-intercept, we set x = 0 in the equation and solve for y. This is because, at any point on the y-axis, the x-coordinate is always 0. Here’s the process:
Given equation: 2x - 3y = 9
Step 1: Substitute x = 0
2(0) - 3y = 9
Step 2: Simplify the equation
0 - 3y = 9
-3y = 9
Step 3: Solve for y
To isolate y, we divide both sides of the equation by -3:
y = 9 / -3
y = -3
Therefore, the y-intercept is the point where the line crosses the y-axis, and its coordinates are (0, -3).
Detailed Explanation of Each Step
Step 1: Substitute x = 0
The first step in determining the y-intercept is to set the x-value to zero. The y-intercept is the point where the line crosses the y-axis, and at any point on this axis, the x-coordinate is always zero. By substituting x with 0, we can find the y-value that satisfies the equation at this specific point.
In the equation 2x - 3y = 9, we replace x with 0:
2(0) - 3y = 9
This substitution allows us to simplify the equation and solve for y.
Step 2: Simplify the Equation
After substituting x = 0, the equation becomes:
2(0) - 3y = 9
Next, simplify the equation by performing the multiplication:
0 - 3y = 9
Since adding or subtracting 0 does not change the value, the equation is simplified to:
-3y = 9
This simplified form brings us closer to isolating y and determining its value. By eliminating the term involving x, we have an equation that is easily solvable.
Step 3: Solve for y
The final step in finding the y-intercept is to solve for y. The simplified equation is:
-3y = 9
To isolate y, we need to undo the multiplication by -3. We do this by dividing both sides of the equation by -3, which maintains the balance of the equation:
-3y / -3 = 9 / -3
This simplifies to:
y = -3
Thus, the y-coordinate of the y-intercept is -3. Since the x-coordinate is 0 at the y-intercept, the coordinates of the y-intercept are (0, -3).
Summary of Intercepts for 2x - 3y = 9
After following the steps outlined above, we have found the x- and y-intercepts for the equation 2x - 3y = 9:
- x-intercept: (4.5, 0)
- y-intercept: (0, -3)
These two points are critical for graphing the line represented by the equation 2x - 3y = 9. By plotting these points on a coordinate plane and drawing a line through them, you can visualize the entire linear relationship.
Importance of Accuracy
When calculating intercepts, accuracy is paramount. A small mistake in arithmetic can lead to incorrect coordinates, which will result in an incorrect graph. It is always a good practice to double-check your calculations and ensure that the intercepts make sense in the context of the equation.
For example, if you find an x-intercept of (-4.5, 0) instead of (4.5, 0), it indicates a calculation error because substituting these values back into the original equation will not satisfy it.
Graphing the Line
Once you have the intercepts, graphing the line is straightforward. Here’s a step-by-step guide:
- Plot the intercepts: Plot the x-intercept (4.5, 0) and the y-intercept (0, -3) on the coordinate plane.
- Draw the line: Use a straightedge to draw a line that passes through both points. Extend the line beyond the points to show the line's full extent.
- Verify the line: Optionally, you can choose another point on the line by selecting an x-value, substituting it into the equation, and solving for y. Plot this point and ensure it lies on the line you’ve drawn.
Graphing the line provides a visual representation of the equation and allows for a quick check of the intercepts. A correctly drawn line should pass precisely through the calculated intercept points.
Applications in Real-World Scenarios
Understanding how to find intercepts is not just an academic exercise; it has practical applications in various real-world scenarios. Intercepts can provide meaningful information in fields such as economics, physics, and engineering.
Economics
In economics, linear equations are often used to model cost functions, revenue functions, and supply-demand relationships. The intercepts of these functions can have significant economic interpretations. For example:
- Cost Function: If a linear equation represents a cost function, the y-intercept represents the fixed costs (the costs incurred even when production is zero), and the x-intercept can represent the break-even point (the point at which revenue equals costs).
- Supply and Demand: In a supply-demand model, the intercepts can represent the price at which there is no demand (y-intercept) or the quantity supplied when the price is zero (x-intercept).
Physics
In physics, linear equations can describe motion, such as the relationship between distance and time for an object moving at a constant velocity. The intercepts can represent:
- Initial Position: The y-intercept can represent the object's initial position at time t = 0.
- Time of Arrival: The x-intercept can represent the time at which the object reaches a certain position or crosses a specific point.
Engineering
Engineers use linear equations in various applications, including circuit analysis, structural analysis, and control systems. Intercepts can represent:
- Initial Conditions: In control systems, the y-intercept might represent the initial state of a system.
- Threshold Values: In circuit analysis, the intercepts could represent threshold voltages or currents.
By understanding the context of a linear equation, the intercepts can provide valuable insights and help in making informed decisions.
Common Mistakes to Avoid
When finding intercepts, several common mistakes can lead to incorrect results. Being aware of these pitfalls can help you avoid them and ensure accuracy in your calculations.
1. Incorrect Substitution
A common mistake is substituting the wrong value for either x or y. Remember, to find the x-intercept, you set y = 0, and to find the y-intercept, you set x = 0. Mixing these up will lead to incorrect intercepts.
Example:
- Incorrect: To find the x-intercept, set x = 0.
- Correct: To find the x-intercept, set y = 0.
2. Arithmetic Errors
Simple arithmetic errors, such as incorrect addition, subtraction, multiplication, or division, can lead to wrong answers. Always double-check your calculations, especially when dealing with negative numbers or fractions.
Example:
- Incorrect: 2x = 9, so x = 9 / 4 (instead of 9 / 2)
- Correct: 2x = 9, so x = 9 / 2 = 4.5
3. Incorrectly Solving for the Variable
When solving for x or y, it’s crucial to perform the algebraic steps correctly. Make sure to isolate the variable by performing the inverse operations in the correct order.
Example:
- Incorrect: -3y = 9, so y = 9 / 3 (forgetting the negative sign)
- Correct: -3y = 9, so y = 9 / -3 = -3
4. Misinterpreting Intercepts as Coordinates
It’s essential to express intercepts as coordinates (x, y). For the x-intercept, the y-coordinate is always 0, and for the y-intercept, the x-coordinate is always 0. Failing to include the zero can lead to confusion.
Example:
- Incorrect: The x-intercept is 4.5.
- Correct: The x-intercept is (4.5, 0).
5. Not Simplifying Equations Properly
Failing to simplify equations before solving can lead to unnecessary complexity and errors. Always simplify the equation as much as possible before isolating the variable.
Example:
- Equation:* 2(0) - 3y = 9
- Incorrect simplification: 2 - 3y = 9
- Correct simplification: 0 - 3y = 9
By avoiding these common mistakes, you can improve your accuracy and confidence in finding intercepts of linear equations.
Conclusion
Finding the x- and y-intercepts of a linear equation is a fundamental skill in mathematics with wide-ranging applications. In this guide, we've walked through the process of finding these intercepts for the equation 2x - 3y = 9. By setting y = 0 to find the x-intercept and x = 0 to find the y-intercept, we identified the points where the line crosses the axes. The x-intercept was found to be (4.5, 0), and the y-intercept was (0, -3).
These intercepts are not just points on a graph; they provide critical information about the linear relationship being represented. They can be used to graph the line, understand real-world applications, and interpret the behavior of the equation. By avoiding common mistakes and practicing these techniques, you can confidently find intercepts for any linear equation, enhancing your problem-solving skills in mathematics and beyond. Whether you're plotting graphs, analyzing economic functions, or solving physics problems, the ability to find intercepts accurately is a valuable asset.
