Solving Linear Equations: Find The Solution Pair

Solving Linear Equations: Find the Solution Pair

Hey guys! Ever stared at a system of linear equations and wondered which ordered pair is the golden ticket to the solution? Today, we're diving deep into a problem that'll have you flexing those math muscles and finding that elusive $(a, b)$ pair. We've got ourselves a classic system:

$\left\{\begin{aligned} 3 a+b & =10 \\ -4 a-2 b & =2 \end{aligned}\right.$

Our mission, should we choose to accept it (and we totally should, because math is awesome!), is to figure out which of the following ordered pairs is the true solution:

A. $(1,7)$ B. $(3,1)$ C. $(11,-23)$ D. $(23,-11)$

Let's get this party started!

The Substitution Strategy: A Reliable Friend

One of the most straightforward ways to tackle this, my friends, is the substitution method. The goal here is to isolate one variable in one of the equations and then plug that expression into the other equation. Looking at our system, the first equation, $3a + b = 10$, looks super friendly for this. We can easily get $b$ by itself.

Isolate $b$ from the first equation:

$3a + b = 10$ Subtract $3a$ from both sides: $b = 10 - 3a$

Now, boom! We have an expression for $b$. The next step is to substitute this expression ($10 - 3a$) for $b$ in the second equation: $-4a - 2b = 2$.

Substitute and solve for $a$:

$-4a - 2(10 - 3a) = 2$

Distribute the $-2$ across the terms inside the parentheses:

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

Now, let's combine the like terms (the $a$ terms):

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

$2a - 20 = 2$

Time to get $a$ all by its lonesome. Add $20$ to both sides:

$2a = 2 + 20$

$2a = 22$

Finally, divide by $2$ to find the value of $a$:

$a = \frac{22}{2}$

$a = 11$

See? We found our $a$ value! Now, before we get too excited, we need to find the corresponding $b$ value. Remember that awesome expression we got for $b$ earlier? $b = 10 - 3a$. Let's plug our newly found $a=11$ back into that.

Solve for $b$:

$b = 10 - 3(11)$

$b = 10 - 33$

$b = -23$

So, the ordered pair solution we've found is $(11, -23)$.

The Elimination Method: A Powerful Alternative

Alright guys, let's say substitution isn't your jam, or maybe you just want to see another way to conquer this beast. The elimination method is another super effective strategy. The idea here is to manipulate one or both equations so that when you add or subtract them, one of the variables cancels out.

Let's look at our system again:

$\left\{\begin{aligned} 3 a+b & =10 \\ -4 a-2 b & =2 \end{aligned}\right.$

Notice the coefficients of $b$. In the first equation, we have $1b$, and in the second, we have $-2b$. If we multiply the entire first equation by $2$, we'll get $2b$ in the first equation, which will perfectly cancel out the $-2b$ in the second equation when we add them.

Multiply the first equation by 2:

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

$6a + 2b = 20$

Now, let's rewrite our system with this modified first equation:

$\left\{\begin{aligned} 6 a+2 b & =20 \\ -4 a-2 b & =2 \end{aligned}\right.$

Look at that! The $2b$ and $-2b$ terms are ready to vanish. Let's add the two equations together:

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

Combine like terms:

$(6a - 4a) + (2b - 2b) = 22$

$2a + 0 = 22$

$2a = 22$

And just like before, we solve for $a$ by dividing by $2$:

$a = \frac{22}{2}$

$a = 11$

We got $a=11$ again! Awesome. Now, we need to find $b$. We can substitute this value of $a$ back into either of the original equations. Let's use the first one, $3a + b = 10$, because it seems a bit simpler.

Substitute $a=11$ into the first original equation:

$3(11) + b = 10$

$33 + b = 10$

To isolate $b$, subtract $33$ from both sides:

$b = 10 - 33$

$b = -23$

And there you have it! The solution is indeed $(11, -23)$.

Checking Our Work: The Ultimate Confidence Booster

Now, here's the crucial part, guys: always check your answer! It's the best way to ensure you haven't made any silly mistakes and that your ordered pair is truly the solution. We need to plug our solution $(11, -23)$ into both original equations to make sure they hold true.

Check in the first equation: $3a + b = 10$

Substitute $a=11$ and $b=-23$:

$3(11) + (-23) = 10$

$33 - 23 = 10$

$10 = 10$

Success! The first equation checks out.

Check in the second equation: $-4a - 2b = 2$

Substitute $a=11$ and $b=-23$:

$-4(11) - 2(-23) = 2$

$-44 - (-46) = 2$

$-44 + 46 = 2$

$2 = 2$

Double success! The second equation also checks out. This confirms that our ordered pair $(11, -23)$ is the correct solution to the system of linear equations.

Analyzing the Options: Why the Others Don't Make the Cut

We found our solution is $(11, -23)$, which corresponds to option C. But let's quickly look at why the other options wouldn't work, just for kicks and to solidify our understanding. It's all about substitution!

Option A: $(1, 7)$

Let's plug $a=1$ and $b=7$ into the first equation: $3a + b = 10$ $3(1) + 7 = 3 + 7 = 10$. This equation works!

Now, let's check the second equation: $-4a - 2b = 2$ $-4(1) - 2(7) = -4 - 14 = -18$. $-18 \neq 2$. So, $(1,7)$ is not the solution.

Option B: $(3, 1)$

Check the first equation: $3a + b = 10$ $3(3) + 1 = 9 + 1 = 10$. This one works too!

Check the second equation: $-4a - 2b = 2$ $-4(3) - 2(1) = -12 - 2 = -14$. $-14 \neq 2$. So, $(3,1)$ is not the solution.

Option D: $(23, -11)$

Check the first equation: $3a + b = 10$ $3(23) + (-11) = 69 - 11 = 58$. $58 \neq 10$. This doesn't even work for the first equation! So, $(23, -11)$ is definitely not the solution.

As you can see, only our answer $(11, -23)$ satisfies both equations. So, the correct answer is C.

Final Thoughts

Mastering systems of linear equations is a fundamental skill in mathematics, guys. Whether you prefer the elegance of substitution or the directness of elimination, the key is practice and careful calculation. Remember to always check your answers to ensure accuracy. Keep those math skills sharp, and you'll be solving complex problems like this one in no time! Happy calculating!