Solving A System Of Linear Equations: Find (a, B)

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Hey guys! Today, we're diving into a fun little math problem: finding the solution to a system of linear equations. Specifically, we need to determine which ordered pair (a, b) satisfies both equations in the following system:

3a + b = 10
-4a - 2b = 2

Let's explore this problem and find the correct answer. This is an important skill in algebra, and I'm sure you'll find it useful!

Understanding the Problem

Before we jump into solving, let's make sure we understand what the question is asking. We are given two linear equations with two variables, a and b. Our goal is to find a pair of values for a and b that, when plugged into both equations, make both equations true. Basically, we're looking for the point where these two lines intersect if we were to graph them. The provided options are ordered pairs, and we need to pick the one that works.

Why This Matters

Solving systems of equations is a fundamental skill in mathematics and has applications in various real-world scenarios. From determining the break-even point in business to modeling physical systems, the ability to find solutions to multiple equations simultaneously is incredibly useful. Understanding the underlying concepts will not only help you ace your math tests but also equip you with problem-solving skills applicable in many different fields.

Review of Linear Equations

Let's briefly recap what linear equations are. A linear equation is an equation that can be written in the form Ax + By = C, where A, B, and C are constants, and x and y are variables. When graphed on a coordinate plane, a linear equation represents a straight line. The solution to a system of linear equations is the point (or points) where the lines intersect. If the lines are parallel, there is no solution; if they are the same line, there are infinitely many solutions. In our case, we are assuming there is a unique solution, which is one ordered pair. We need to find it.

Methods to Solve Systems of Equations

There are a few common methods to solve systems of linear equations. Let's quickly review them:

  1. Substitution: Solve one equation for one variable, and then substitute that expression into the other equation.
  2. Elimination (or Addition/Subtraction): Multiply one or both equations by constants so that the coefficients of one of the variables are opposites. Then, add the equations together to eliminate that variable.
  3. Graphing: Graph both equations on the same coordinate plane and find the point of intersection.

For this problem, the elimination method seems like a good approach, so let's use that one.

Solving the System of Equations Using Elimination

Our system of equations is:

3a + b = 10
-4a - 2b = 2

To use the elimination method, we want to eliminate either a or b. Let's eliminate b. To do this, we can multiply the first equation by 2:

2 * (3a + b) = 2 * 10

This gives us:

6a + 2b = 20

Now we have the following system:

6a + 2b = 20
-4a - 2b = 2

Notice that the coefficients of b are now opposites (2 and -2). We can add the two equations together to eliminate b:

(6a + 2b) + (-4a - 2b) = 20 + 2

This simplifies to:

2a = 22

Now we can solve for a:

a = 22 / 2
a = 11

Great! We found that a = 11. Now we can substitute this value back into one of the original equations to solve for b. Let's use the first equation:

3a + b = 10
3(11) + b = 10
33 + b = 10

Subtract 33 from both sides:

b = 10 - 33
b = -23

So, we found that b = -23.

Checking the Solution

It's always a good idea to check our solution by plugging the values of a and b back into both original equations to make sure they are satisfied. Let's do that:

Equation 1: 3a + b = 10

3(11) + (-23) = 33 - 23 = 10  (Correct!)

Equation 2: -4a - 2b = 2

-4(11) - 2(-23) = -44 + 46 = 2  (Correct!)

Since our values for a and b satisfy both equations, we know our solution is correct.

Identifying the Correct Ordered Pair

We found that a = 11 and b = -23. Therefore, the ordered pair (a, b) is (11, -23).

Looking at the answer choices, we see that option C, (11, -23), is the correct answer.

Conclusion

By using the elimination method, we successfully solved the system of linear equations and found the ordered pair that satisfies both equations. Remember, it's crucial to understand the underlying principles and be comfortable with the different methods to solve these types of problems. Keep practicing, and you'll become a pro at solving systems of equations!

So the final answer is C. (11, -23).

Key Takeaways:

  • Systems of equations can be solved using substitution, elimination, or graphing.
  • The elimination method involves manipulating equations to eliminate one variable.
  • Always check your solution by plugging the values back into the original equations.
  • Understanding the basics is important, so keep practicing!

I hope this explanation was helpful! Keep practicing, and you'll master these types of problems in no time. Good luck, and happy solving!