Here's how to solve 5x + 7 = 22 and find x in linear equations.

How to Crack an Equation Like a Pro (And Why It Helps on the HSPT)

Let me ask you a quick question. Have you ever looked at a simple equation and felt like you were staring at a tiny riddle? Sometimes the math looks tougher than it is, until you spot the easy moves hiding inside. Today we’ll use a clean, everyday example to show how to solve for x—and why that approach shows up again and again on the HSPT math topics you’ll encounter.

A tiny puzzle with a tidy answer

Here’s a classic, straightforward problem you might see in the math section:

If 5x + 7 = 22, what is x?

A. 2

B. 3

C. 4

D. 5

The right choice is 3. Let’s walk through how we get there, step by step.

Step 1: Remove the constant piece from the left

To see the x clearly, you want to isolate the term that has x. That means getting rid of the extra 7 that sits on the left side. You do this by doing the same thing to both sides of the equation—subtract 7 from each side:

5x + 7 − 7 = 22 − 7

That simplifies to:

5x = 15

Nice and simple. No sneaky tricks, just keeping the equation balanced.

Step 2: Make the x stand alone

Now you have 5x on the left. The only thing left is to remove the 5 that’s multiplying x. You do that by dividing both sides by 5:

(5x) / 5 = 15 / 5

So x = 3.

That’s all there is to it. The method—subtract then divide—is the standard way to squeeze the unknown out of the left-hand side while keeping the equation in balance.

Double-checking is a friendly habit

A quick check helps your confidence. Plug x = 3 back into the original equation:

5(3) + 7 = 15 + 7 = 22

Both sides match. If they don’t, you know something went off. A good habit is to test your solution, especially with linear equations like this one.

Why this approach makes sense

What you did is a clean example of inverse operations in action. Subtraction undid the constant term (the 7) on the left, and division undid the multiplication by 5. The goal is to get the variable all by itself, so you can see its value clearly.

In the HSPT math topics, you’ll see many problems that rely on this same idea—solving for x, a variable, or sometimes another letter in simple equations. The move is universal: identify what’s preventing the variable from being alone, and reverse the steps that built up that barrier.

Common traps and how to avoid them

Even when the math is simple, it’s easy to stumble. Here are a few things to watch out for, along with quick fixes:

• Forgetting to apply the operation to both sides

If you only subtract (or only divide) on one side, you break the balance. Always perform the same operation on both sides.

• Dropping a sign

Minus signs sneak in and out more than you expect. If you’re subtracting, keep an eye on whether a negative or a positive result is really intended.

• Jumping to conclusions

Sometimes there are several easy steps that lead to the answer, and you might skip one. Slow down enough to trace each move. A short recap helps.

• Not checking the work

If you don’t test the answer, you might miss a small arithmetic slip. A quick substitution is worth it.

What this tells us about math on the test

The HSPT math topics often test your facility with basic algebra because it’s the groundwork for more involved problems—geometry walls, word problems, even data tables. When you can confidently isolate a variable, you’re building a sturdy toolkit. It’s less about memorizing a trick and more about knowing what to do when a variable is playing hide-and-seek.

A little more practice, a little more clarity

If you like, try another simple equation to feel the rhythm:

4y − 9 = 7

First, add 9 to both sides (the inverse of subtraction):

4y = 16

Then divide by 4:

y = 4

See? The same two moves, just in a different order of operations. The pattern becomes familiar, and familiarity brings speed and accuracy—two big advantages on any test.

Relating math to everyday life

This isn’t just a mental exercise. The idea of isolating a variable crops up in real life, too. Think about budgeting a small trip. If you know you’ll spend a fixed amount on transport and food, you might set up a simple equation to figure out how much you can spend on something else, given a total. Or picture a recipe where you’re adjusting a single ingredient to keep the flavor balanced. You’re essentially solving for that unknown amount so the whole dish—your budget or your meal—turns out just right.

In the same vein, word problems on the HSPT often disguise a variable in a real-sounding scenario. The more comfortable you are with the algebra, the easier it is to turn the story into a clean equation you can solve in your head or on paper.

Strategies that help, without the fluff

Here are practical tips you can tuck into your study habits (without turning into a big drill sergeant routine):

• Keep operations paired

Always perform the same operation on both sides. If you add, you add to both sides; if you subtract, you subtract from both sides. It’s the easiest way to keep things honest.

• Use a quick substitute check

After you solve for x, pick a simple value and plug it back. If the equation balances, you’re likely on the right track.

• Write the steps cleanly

Even a tiny scratch pad note helps prevent missteps. A neat sequence—subtract, then divide—serves as a reliable roadmap.

• Don’t race to the answer

Speed matters, but accuracy matters more. If you feel yourself rushing, slow down. A deliberate pace today saves you trouble later.

• See the pattern, then generalize

If you can identify the two-step pattern in one problem, you can apply it to many others. Turning a tiny insight into a general habit pays off on the test and beyond.

A deeper look at the math language

There’s a small, almost subtle, language to the math you’re learning: “the left side,” “the right side,” “the variable,” “the constant.” The words aren’t just flavor; they guide your actions. When you say “isolate the variable,” you’re setting a goal. When you say “subtract 7 from both sides,” you’re naming a precise move. The clarity of language helps focus your eyes and your brain. On tests like the HSPT, that clarity can shave seconds off your work while reducing careless mistakes.

A few optional challenges to sharpen your sense

If you want to push a little further without getting overwhelmed, try these quick twists:

• Solve 7x + 4 = 31

Subtract 4: 7x = 27; divide by 7: x = 27/7 (about 3.857). It’s a good reminder that not all answers are whole numbers, and that’s perfectly fine in algebra.

• Solve −2z + 9 = 1

Subtract 9: −2z = −8; divide by −2: z = 4. A reminder that negative signs can flip the direction of the operation.

• Solve 3(a − 5) = 12

First, divide both sides by 3: a − 5 = 4; then add 5: a = 9. This shows you can spread the work across parentheses and still keep the core idea intact.

Bringing it back to the bigger picture

So, what’s the big takeaway? When you face a linear equation, the path to the answer is usually two simple moves: undo what’s built up around the variable, then check your result. It sounds almost too small to matter, but it’s a little method that pays off with consistency and calm, especially in a testing situation.

On the HSPT math topics, you’ll likely encounter more equations like this, sometimes with a few extra layers of steps or a short word problem wrapped around the math. The core skill stays the same: identify what you need to undo to reveal the unknown, apply the inverse operation, and verify.

A final nudge of encouragement

If you’re reading this and thinking, “This makes sense,” you’re already tilting the odds in your favor. Math is less about memorizing a parade of tricks and more about recognizing a reliable rhythm—identify, isolate, verify. The better you recognize that rhythm, the more confident you’ll feel when you see a fresh problem.

And yes, that calm confidence tends to spill over into other parts of school and even daily life. When you can break a puzzle into a couple of clean moves, you feel more in control, not just with numbers but with problems in general. That’s a nice side effect of learning how to solve equations: you gain a practical superpower for real-world thinking.

If you’re curious, keep exploring simple equations and their twins. Try siblings of the same structure, switch the numbers around, and test yourself with friendly challenges. The more you practice that two-step pattern, the more natural it becomes. Before long, you’ll notice your speed and accuracy improving—without needing a dramatic overhaul of how you study.

In the end, a small equation like 5x + 7 = 22 isn’t just a math note. It’s a reminder that the right moves—subtract, then divide—can turn a puzzle into a clear answer. And that clarity, once it lands, travels with you through all kinds of problems, on tests and in everyday life.