Solve 3x + 2y = 10 & 4x - Y = 6: A Step-by-Step Guide
Hey guys! Today, we're diving deep into the world of linear equations and system-solving. Specifically, we're going to tackle the system:
3x + 2y = 10
4x - y = 6
This might look intimidating at first, but don't worry! We'll break it down step-by-step, exploring different methods to find the values of 'x' and 'y' that satisfy both equations. Whether you're a student grappling with algebra or just someone looking to refresh your math skills, this guide is for you. So, let's jump right in and unravel the mysteries of these equations!
Understanding Systems of Equations
Before we start crunching numbers, let's get a handle on what a system of equations actually is. Imagine you have two or more equations, each with two or more variables (like 'x' and 'y'). A system of equations is simply a set of these equations considered together. The goal? To find the values for the variables that make all the equations true at the same time. Think of it as a puzzle where each equation is a clue, and you need to find the solution that fits all the clues perfectly.
In our case, we have two equations:
- 3x + 2y = 10
- 4x - y = 6
Each of these equations represents a straight line when graphed on a coordinate plane. The solution to the system is the point where these lines intersect. This point (x, y) is the one that satisfies both equations simultaneously. There are several methods we can use to find this magical point of intersection, and we'll explore a few of the most common ones.
Method 1: The Substitution Method
The substitution method is a powerful technique for solving systems of equations. The basic idea is to solve one equation for one variable and then substitute that expression into the other equation. This eliminates one variable, leaving you with a single equation in a single variable, which is much easier to solve. Let's see how it works with our system.
Step 1: Solve one equation for one variable.
Looking at our equations, the second one (4x - y = 6) seems easier to manipulate. Let's solve it for 'y'.
Add 'y' to both sides: 4x = y + 6 Subtract 6 from both sides: 4x - 6 = y
So, we have y = 4x - 6. Great! We've isolated 'y'.
Step 2: Substitute the expression into the other equation.
Now, we'll take this expression for 'y' (4x - 6) and substitute it into the first equation (3x + 2y = 10). This is crucial â we substitute into the equation we didn't use in step 1. This prevents us from getting stuck in a loop.
So, 3x + 2(4x - 6) = 10
Step 3: Solve the resulting equation.
Now we have an equation with only 'x', which we can solve using basic algebra.
Distribute the 2: 3x + 8x - 12 = 10 Combine like terms: 11x - 12 = 10 Add 12 to both sides: 11x = 22 Divide both sides by 11: x = 2
Fantastic! We've found the value of 'x': x = 2.
Step 4: Substitute the value back to find the other variable.
Now that we know x = 2, we can substitute it back into either of our original equations (or the equation we solved for 'y' in step 1) to find 'y'. Let's use the equation y = 4x - 6, since we already have 'y' isolated.
y = 4(2) - 6 y = 8 - 6 y = 2
So, we've found the value of 'y': y = 2.
Step 5: Check your solution.
It's always a good idea to check your solution by plugging the values of 'x' and 'y' back into both original equations to make sure they hold true.
Equation 1: 3x + 2y = 10 3(2) + 2(2) = 6 + 4 = 10 (Correct!)
Equation 2: 4x - y = 6 4(2) - 2 = 8 - 2 = 6 (Correct!)
Since our solution (x = 2, y = 2) satisfies both equations, we know we've found the correct answer.
Method 2: The Elimination Method
The elimination method, also known as the addition method, provides another powerful way to solve systems of equations. This method focuses on eliminating one of the variables by manipulating the equations so that the coefficients of one variable are opposites. When you add the equations together, that variable cancels out, leaving you with a single equation in a single variable. Let's see how this works with our system:
Step 1: Manipulate the equations to get opposite coefficients for one variable.
Looking at our system:
- 3x + 2y = 10
- 4x - y = 6
The coefficients of 'y' are 2 and -1. We can easily make these opposites by multiplying the second equation by 2.
Multiply the second equation by 2: 2(4x - y) = 2(6) => 8x - 2y = 12
Now our system looks like this:
- 3x + 2y = 10
- 8x - 2y = 12
Notice that the coefficients of 'y' are now 2 and -2, which are opposites.
Step 2: Add the equations together.
Now we add the two equations together, term by term:
(3x + 2y) + (8x - 2y) = 10 + 12
Combine like terms: 11x = 22
Notice how the 'y' terms canceled out, as planned!
Step 3: Solve the resulting equation.
We're left with a simple equation in 'x':
11x = 22
Divide both sides by 11: x = 2
We've found the value of 'x': x = 2. This is the same value we found using the substitution method, which is a good sign!
Step 4: Substitute the value back to find the other variable.
Just like with the substitution method, we now substitute the value of 'x' (x = 2) back into either of the original equations to find 'y'. Let's use the first equation:
3x + 2y = 10 3(2) + 2y = 10 6 + 2y = 10 Subtract 6 from both sides: 2y = 4 Divide both sides by 2: y = 2
We've found the value of 'y': y = 2. Again, this matches the result we obtained with the substitution method.
Step 5: Check your solution.
As always, we check our solution by plugging the values of 'x' and 'y' back into both original equations:
Equation 1: 3x + 2y = 10 3(2) + 2(2) = 6 + 4 = 10 (Correct!)
Equation 2: 4x - y = 6 4(2) - 2 = 8 - 2 = 6 (Correct!)
Our solution (x = 2, y = 2) satisfies both equations, confirming that we've solved the system correctly.
Method 3: Graphical Solution
While the substitution and elimination methods are algebraic techniques, we can also solve systems of equations graphically. Each equation in the system represents a line, and the solution to the system is the point where the lines intersect. This method provides a visual representation of the solution.
Step 1: Rewrite each equation in slope-intercept form (y = mx + b).
This form makes it easy to identify the slope (m) and y-intercept (b) of each line, which we need for graphing.
Equation 1: 3x + 2y = 10 Subtract 3x from both sides: 2y = -3x + 10 Divide both sides by 2: y = (-3/2)x + 5
So, the first equation has a slope of -3/2 and a y-intercept of 5.
Equation 2: 4x - y = 6 Subtract 4x from both sides: -y = -4x + 6 Multiply both sides by -1: y = 4x - 6
The second equation has a slope of 4 and a y-intercept of -6.
Step 2: Graph both lines on the same coordinate plane.
To graph each line, you can use the slope and y-intercept or find two points that lie on the line. For example:
- For the first line (y = (-3/2)x + 5), the y-intercept is 5, so we have the point (0, 5). We can also find another point by plugging in x = 2: y = (-3/2)(2) + 5 = 2, so we have the point (2, 2).
- For the second line (y = 4x - 6), the y-intercept is -6, so we have the point (0, -6). Plugging in x = 1 gives y = 4(1) - 6 = -2, so we have the point (1, -2).
Plot these points and draw the lines.
Step 3: Identify the point of intersection.
The point where the two lines intersect is the solution to the system of equations. If you graph the lines accurately, you'll see that they intersect at the point (2, 2).
Step 4: Check your solution.
As with the other methods, it's essential to check your solution by plugging the values of 'x' and 'y' back into the original equations. We've already done this in the previous methods, and we know that (2, 2) satisfies both equations.
Conclusion
We've explored three different methods for solving the system of equations:
3x + 2y = 10
4x - y = 6
We found that the solution is x = 2 and y = 2, or the point (2, 2). Whether you prefer the algebraic precision of the substitution and elimination methods or the visual clarity of the graphical method, you now have the tools to tackle similar systems of equations with confidence.
Key Takeaways:
- Systems of equations represent a set of equations considered together.
- The solution to a system is the set of values for the variables that satisfy all equations simultaneously.
- The substitution method involves solving one equation for one variable and substituting that expression into the other equation.
- The elimination method involves manipulating the equations to eliminate one variable by adding the equations together.
- The graphical method involves graphing the equations and finding the point of intersection.
So there you have it, guys! You've conquered another mathematical challenge. Keep practicing, and you'll become a system-solving pro in no time!
