Finding Solutions: Understanding Algebraic Equations

How many solutions exist for the equations x - 2 = 5 and 2x + 3 = 15?

• A. Zero
• B. One
• C. Two
• D. Infinite

Answer: (B)

The equation \( x - 2 = 5 \) can be solved by isolating \( x \). Adding 2 to both sides gives: \[ x = 5 + 2 = 7 \] This means that there is one solution for the equation \( x - 2 = 5 \): \( x = 7 \). Next, for the equation \( 2x + 3 = 15 \), we can isolate \( x \) by first subtracting 3 from both sides: \[ 2x = 15 - 3 = 12 \] Now, dividing both sides by 2 results in: \[ x = \frac{12}{2} = 6 \] This indicates that there is also one solution for the equation \( 2x + 3 = 15 \): \( x = 6 \). Since each equation yields a single, unique solution, the overall conclusion is that both equations have one solution each, but they do not intersect or yield the same solution. The question is asking about the total number of solutions across the two equations, and each equation is independently true for the value it provides. Hence, the answer is that there is one solution

When you're faced with equations like ( x - 2 = 5 ) and ( 2x + 3 = 15 ), you might feel a twinge of anxiety, right? But fear not! Unlocking these algebraic mysteries is easier than you think. Let's unravel this together, shall we?

Imagine you're on a treasure hunt for the value of ( x ). The first equation, ( x - 2 = 5 ), is your first clue. To find ( x ), you just need to isolate it. If you add 2 to both sides, what do you get? That's right!

[

x = 5 + 2 = 7

]

Ta-da! You've found ( x = 7 ). But hang on; that’s just one solution. This equation isn’t going to throw any curveballs at you; it leads you directly to only one answer.

Now, let's shift gears to our second equation: ( 2x + 3 = 15 ). At first glance, it might look a bit tricky, but it’s simpler than it appears. Start by subtracting 3 from both sides to isolate the term with ( x ):

[

2x = 15 - 3 = 12

]

Next, divide by 2 to find ( x ). You got it!

[

x = \frac{12}{2} = 6

]

Boom! Another unique solution: ( x = 6 ). But here’s the kicker—while both equations have individual solutions, they don’t share the same ( x ) value. So, what’s the grand takeaway?

Both equations yield exactly one solution each. If you were asked how many solutions exist for the combined equations, the simple answer is: just one, per equation!

Now, let’s reflect on that: why is it important to grasp these concepts? It’s because understanding how to manipulate equations is not just about passing a test; it’s about developing critical thinking. You see, equations are staircases leading to higher levels of math and beyond. When you tackle these problems and come out the other side, you build confidence.

So, what’s next? Practice makes perfect. Try more problems that involve isolating variables, or even mix it up by adding some complexity, like working with systems of equations. You know what? The more you practice, the more those numbers will start to tell a story, one you’ll be fluent in.

So go ahead, embrace the equations! The solutions are waiting for you. Need more examples? Ask a teacher or look for resources online. Remember: math isn’t just about the right answer—it’s about the journey to get there. Happy solving!