Hey guys! Today, we're diving into a fundamental algebra problem: solving for a specific variable in a linear equation. Specifically, we'll tackle the equation 7x + 8y = 5 and isolate y. This is a crucial skill in algebra, as it allows us to rewrite equations in different forms, making them easier to analyze, graph, or use in further calculations. So, let’s break down the steps and get you comfortable with this process.
Understanding the Basics of Solving Equations
Before we jump into the specifics of our equation, let's quickly review the core principles behind solving equations. The main idea is to isolate the variable we want to solve for – in this case, y – on one side of the equation. We do this by performing the same operations on both sides of the equation to maintain balance. Think of an equation like a seesaw; if you add or subtract something on one side, you need to do the same on the other to keep it level. The operations we commonly use include addition, subtraction, multiplication, and division. Our goal is to undo the operations that are currently affecting y until it stands alone on one side of the equal sign. Remember, the order of operations (PEMDAS/BODMAS) is crucial in reverse when solving equations. We typically deal with addition and subtraction first, then multiplication and division.
Step 1: Isolate the Term with y
Our starting equation is 7x + 8y = 5. The first step is to isolate the term containing y, which is 8y. To do this, we need to get rid of the 7x term on the left side. Since 7x is being added to 8y, we can undo this addition by subtracting 7x from both sides of the equation. This gives us:
7x + 8y - 7x = 5 - 7x
Simplifying this, we get:
8y = 5 - 7x
Great! Now we have the term with y isolated on the left side. It’s important to perform the same operation on both sides to maintain the equation's balance. Subtracting 7x from both sides ensures that the equation remains true. This step is a cornerstone of solving equations, as it helps us methodically peel away the layers of operations affecting the variable we want to isolate.
Step 2: Isolate y by Dividing
Now that we have 8y = 5 - 7x, the next step is to isolate y completely. Currently, y is being multiplied by 8. To undo this multiplication, we need to divide both sides of the equation by 8. This gives us:
(8y) / 8 = (5 - 7x) / 8
On the left side, the 8 in the numerator and the 8 in the denominator cancel each other out, leaving us with just y. On the right side, we divide the entire expression (5 - 7x) by 8. This is an important point: we are dividing the entire side by 8, not just a part of it. This gives us:
y = (5 - 7x) / 8
We have now successfully isolated y! This division step is crucial because it undoes the multiplication that was binding y, allowing us to express y in terms of x. Remember, dividing both sides by the same non-zero number maintains the equation's balance and is a fundamental algebraic operation.
Final Solution and Different Forms
So, we've found that y = (5 - 7x) / 8 is the solution. This is a perfectly valid way to express y in terms of x. However, sometimes it's useful to rewrite the solution in a slightly different form. We can distribute the division by 8 to both terms in the numerator:
y = 5/8 - (7x)/8
This can also be written as:
y = -7/8x + 5/8
This form is known as the slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept. In our case, the slope is -7/8 and the y-intercept is 5/8. This form is particularly useful when graphing the equation because it directly gives us the slope and y-intercept, making it easy to plot the line. Understanding different forms of the solution can be incredibly beneficial for various applications.
Why is Solving for y Important?
Solving for y is a fundamental skill in algebra with numerous applications. One of the most significant is in graphing linear equations. When an equation is in slope-intercept form, it's incredibly easy to graph. We can identify the slope and y-intercept directly from the equation, allowing us to plot points and draw the line. This is a visual representation of the relationship between x and y, which can be extremely helpful in understanding the equation's behavior.
Beyond graphing, solving for y is crucial in various mathematical and real-world scenarios. For instance, in systems of equations, we often need to solve one equation for one variable to substitute it into another equation. This is a key technique for finding the solution that satisfies both equations simultaneously. In physics and engineering, equations often describe relationships between different variables, and solving for a particular variable allows us to make predictions and analyze systems. For example, we might need to solve for the distance an object travels given its speed and time, or calculate the force required to move an object with a certain acceleration.
Common Mistakes to Avoid
When solving equations, it's easy to make mistakes if you're not careful. One common mistake is not performing the same operation on both sides of the equation. Remember, the equation is like a balance, and any operation you perform must be applied equally to both sides to maintain that balance. Another common mistake is incorrectly applying the order of operations. When solving for a variable, we need to reverse the order of operations (PEMDAS/BODMAS), dealing with addition and subtraction before multiplication and division.
For example, in our equation 7x + 8y = 5, a mistake would be to divide only 8y by 8 and not the entire right side. This would lead to an incorrect solution. Another mistake could be subtracting 7x from the left side but adding it to the right side, which would also break the balance of the equation. It’s essential to be meticulous and double-check each step to ensure accuracy. Practicing with different equations and paying attention to these common pitfalls will help you avoid errors and build confidence in your problem-solving abilities.
Practice Problems
To solidify your understanding, let's try a few practice problems. Remember to follow the steps we've discussed: isolate the term with y and then isolate y itself.
- Solve for y: 3x + 2y = 7
- Solve for y: -5x + 4y = 9
- Solve for y: 2x - 6y = 12
Working through these problems will help you internalize the process and recognize patterns that can make solving similar equations easier in the future. Don't be afraid to make mistakes; they are a natural part of the learning process. The key is to learn from them and refine your approach.
Problem 1: 3x + 2y = 7
To solve for y, first, we need to isolate the term containing y, which is 2y. We can do this by subtracting 3x from both sides of the equation:
3x + 2y - 3x = 7 - 3x
This simplifies to:
2y = 7 - 3x
Next, we isolate y by dividing both sides by 2:
(2y) / 2 = (7 - 3x) / 2
This gives us:
y = (7 - 3x) / 2
We can also write this as:
y = 7/2 - (3x)/2
Or in slope-intercept form:
y = -3/2x + 7/2
Problem 2: -5x + 4y = 9
First, we isolate the term with y by adding 5x to both sides:
-5x + 4y + 5x = 9 + 5x
This simplifies to:
4y = 9 + 5x
Now, we divide both sides by 4 to isolate y:
(4y) / 4 = (9 + 5x) / 4
This gives us:
y = (9 + 5x) / 4
We can also write this as:
y = 9/4 + (5x)/4
Or in slope-intercept form:
y = 5/4x + 9/4
Problem 3: 2x - 6y = 12
To isolate the term with y, we subtract 2x from both sides:
2x - 6y - 2x = 12 - 2x
This simplifies to:
-6y = 12 - 2x
Now, we divide both sides by -6 to isolate y:
(-6y) / -6 = (12 - 2x) / -6
This gives us:
y = (12 - 2x) / -6
We can simplify this by dividing each term in the numerator by -6:
y = 12/-6 - (2x)/-6
y = -2 + x/3
Or in slope-intercept form:
y = 1/3x - 2
Conclusion
So, there you have it! We've successfully solved for y in the equation 7x + 8y = 5, and we've also explored the broader context of solving equations and the importance of this skill in algebra. Remember, the key is to isolate the variable you're solving for by performing the same operations on both sides of the equation. With practice, this process will become second nature. Keep up the great work, guys, and happy problem-solving!