Solving 2x-y=8 And 3x+4y=10 Find X And Y
Hey guys! Today, we're diving into the exciting world of solving systems of linear equations. This is a crucial skill in algebra, and we're going to break down a specific problem step-by-step. Our mission? To find the values of x and y that satisfy the following system of equations:
- 2x - y = 8
- 3x + 4y = 10
Don't worry if it looks intimidating at first. We'll explore different methods to tackle this, making sure you understand the logic and reasoning behind each step. So, grab your pencils and notebooks, and let's get started!
Understanding Systems of Equations
Before we jump into solving, let's make sure we're all on the same page about what a system of equations actually is. Think of it as a set of two or more equations that involve the same variables. In our case, we have two equations, and both involve x and y. The solution to the system is the set of values for the variables that make all equations in the system true simultaneously.
Graphically, each linear equation represents a straight line. The solution to the system is the point where these lines intersect. This point of intersection represents the (x, y) coordinates that satisfy both equations. So, we're essentially looking for the coordinates of this intersection point.
There are several methods we can use to solve systems of equations, including:
- Substitution: Solving one equation for one variable and substituting that expression into the other equation.
- Elimination: Adding or subtracting multiples of the equations to eliminate one variable.
- Graphing: Plotting the lines and finding the point of intersection (though this method is less precise for non-integer solutions).
We'll focus on the substitution and elimination methods in this article, as they're the most commonly used and generally more efficient for algebraic solutions. Choosing the right method often depends on the specific equations in the system. Sometimes, one method might be more straightforward than the other.
Method 1: The Substitution Method
The substitution method is a powerful technique for solving systems of equations. The basic idea is to solve one of the equations for one variable in terms of the other, and then substitute that expression into the other equation. This will leave you with a single equation with a single variable, which you can easily solve. Let's apply this to our system:
- 2x - y = 8
- 3x + 4y = 10
Step 1: Solve one equation for one variable.
Looking at our equations, it seems easiest to solve the first equation for y. Adding y to both sides and subtracting 8 from both sides gives us:
2x - 8 = y
So, we have y = 2x - 8. This expresses y in terms of x. This is a crucial step, as we've now isolated y and can use this expression in our next step.
Step 2: Substitute the expression into the other equation.
Now, we'll substitute this expression for y (2x - 8) into the second equation:
3x + 4(2x - 8) = 10
Notice how we've replaced y with its equivalent expression in terms of x. This is the heart of the substitution method. We've now transformed our system into a single equation with only one variable, x.
Step 3: Solve for the remaining variable.
Let's simplify and solve this equation for x:
3x + 8x - 32 = 10
Combining like terms, we get:
11x - 32 = 10
Adding 32 to both sides:
11x = 42
Dividing both sides by 11:
x = 42/11
Step 4: Substitute the value back to find the other variable.
Now that we have the value of x, we can substitute it back into either of the original equations (or the expression we found for y in step 1) to find y. Let's use the expression y = 2x - 8:
y = 2(42/11) - 8
y = 84/11 - 8
To subtract 8, we need a common denominator:
y = 84/11 - 88/11
y = -4/11
Solution:
So, the solution to the system of equations using the substitution method is:
- x = 42/11
- y = -4/11
Therefore, the point of intersection of the two lines represented by these equations is (42/11, -4/11).
Method 2: The Elimination Method
The elimination method, also known as the addition or subtraction method, provides another powerful approach to solving systems of equations. The core idea behind this method is to manipulate the equations in such a way that when you add or subtract them, one of the variables is eliminated. This leaves you with a single equation in one variable, which you can then solve. Let's see how this works with our system:
- 2x - y = 8
- 3x + 4y = 10
Step 1: Multiply equations to make coefficients match (or be opposites).
To eliminate a variable, we need the coefficients of either x or y to be the same (or opposites). Looking at our equations, it seems easiest to eliminate y. The coefficient of y in the first equation is -1, and in the second equation, it's 4. To make them opposites, we can multiply the first equation by 4:
4(2x - y) = 4(8)
This gives us:
8x - 4y = 32
Now, our system looks like this:
- 8x - 4y = 32
- 3x + 4y = 10
Notice that the coefficients of y are now -4 and 4, which are opposites!
Step 2: Add (or subtract) the equations to eliminate a variable.
Since the coefficients of y are opposites, we can add the two equations together. This will eliminate y:
(8x - 4y) + (3x + 4y) = 32 + 10
Combining like terms, we get:
11x = 42
Step 3: Solve for the remaining variable.
Now, we have a simple equation with just x. Dividing both sides by 11, we get:
x = 42/11
Step 4: Substitute the value back to find the other variable.
Just like in the substitution method, we now substitute the value of x back into either of the original equations to solve for y. Let's use the first original equation:
2*(42/11) - y = 8
84/11 - y = 8
To isolate y, let's subtract 84/11 from both sides:
-y = 8 - 84/11
We need a common denominator to subtract:
-y = 88/11 - 84/11
-y = 4/11
Multiplying both sides by -1, we get:
y = -4/11
Solution:
Using the elimination method, we found the solution to be:
- x = 42/11
- y = -4/11
This matches the solution we found using the substitution method, which is a good sign! It confirms that our calculations are correct.
Comparing the Methods
We've successfully solved the system of equations using both the substitution and elimination methods. Both methods are valid and will lead to the same solution if applied correctly. So, which method is better?
- Substitution: This method is particularly useful when one of the equations is already solved for one variable or can be easily solved for one variable. It involves substituting an expression, which can sometimes lead to more complex algebraic manipulations, especially with fractions.
- Elimination: This method shines when the coefficients of one of the variables are the same or easily made the same (or opposites) by multiplication. It involves adding or subtracting equations, which can be a more straightforward approach in some cases.
In our example, both methods worked well. However, you might find that one method is more efficient or easier to apply depending on the specific equations in the system. The key is to understand the underlying principles of each method and choose the one that seems most convenient for the problem at hand.
Why are Systems of Equations Important?
You might be wondering, "Why are we even learning this?" Well, systems of equations are incredibly useful in many real-world applications. They allow us to model situations with multiple constraints or relationships. Here are just a few examples:
- Mixture problems: Determining the amounts of different ingredients needed to create a mixture with specific properties.
- Distance, rate, and time problems: Solving for unknown speeds, distances, or times when there are multiple moving objects.
- Supply and demand in economics: Finding the equilibrium price and quantity where the supply and demand curves intersect.
- Circuit analysis in electrical engineering: Determining the currents and voltages in different parts of a circuit.
- Computer graphics and game development: Calculating transformations and intersections of objects in 2D and 3D space.
Understanding how to solve systems of equations opens doors to tackling a wide range of practical problems. It's a fundamental skill in mathematics and a valuable tool in many other fields.
Practice Makes Perfect
Solving systems of equations is a skill that improves with practice. The more problems you work through, the more comfortable you'll become with the different methods and the quicker you'll be able to identify the best approach. So, don't be afraid to tackle some practice problems!
You can find plenty of examples in textbooks, online resources, and worksheets. Try working through problems using both the substitution and elimination methods to see which you prefer and to solidify your understanding. Remember to check your answers by substituting the values of x and y back into the original equations to make sure they hold true.
Conclusion
We've successfully navigated the world of systems of equations and found the values of x and y that satisfy the equations 2x - y = 8 and 3x + 4y = 10. We explored both the substitution and elimination methods, highlighting the steps involved and the reasoning behind them. Remember, the solution is x = 42/11 and y = -4/11.
More importantly, we've discussed the importance of systems of equations and their applications in various fields. So, keep practicing, keep exploring, and keep applying your knowledge to solve real-world problems. You've got this!
