Solving Linear Equations

0:00 Welcome! So, you're prepping for the ASVAB and maybe feeling a little uneasy about the math section? Well, you are definitely not alone. I'm so glad you're here for this explainer, because today we are going to completely demystify linear equations. The mathematics knowledge subtest relies heavily on high school algebra principles, specifically moving things around and isolating variables. And here's the kicker. No calculators allowed. Zero. But hey, don't worry, conquering these equations is literally just about following a steady logical step-by-step process with nothing but your pencil and scratch paper. Once you start seeing the patterns, it honestly becomes really satisfying. Here's our clear roadmap for today. We'll start with equation basics and rules, then move on to handling fractions and decimals. After that, word problems and formulas, followed by some step-by-step worked examples. Finally, we'll wrap up with costly mistakes to avoid, and a quick check yourself final practice. It's a logical, totally manageable path. Alright, let's kick things off with section 1, equation basics and rules. Before we start solving anything, we really need to understand exactly what we're looking at.

1:01 Notice the clear distinction here between an expression and an equation. An expression, like 4x minus 9, is basically just a mathematical phrase. It doesn't have an equal sign. You can simplify it, sure, but you can't actually solve it. An equation, on the other hand, makes a bold claim of balance. It says 4x minus 9 is exactly equal to 11. That equal sign? It's everything. It tells us we have a real claim that we can actively solve by finding the specific value of x that makes the whole statement true. Think of an equation like a perfectly balanced scale. Whatever you do to one side, you have got to do to the other to keep that balance. We solve these using inverse operations, which is really just a fancy way of saying we undo the math in reverse order. So take a look at step 1. 4x minus 5 equals 23. To undo that subtraction, we just add 5 to both sides. That brings us nicely to step 2. 4x equals 28. Now to undo the multiplication of 4 times x, we divide both sides by 4.

2:03 And that gives us our final answer right there in step 3. x equals 7. Simple, perfectly balanced, and totally step by step. Handling fractions and decimals. Now, this is where things can look a little intimidating, but seriously, don't let it psych you out. Those scary looking fractions and decimals can be wiped out right at the start. This right here illustrates one of my absolute favorite shortcuts in algebra. When you see multiple fractions, like in step 1 where we have x over 2 plus x over 3 equals 10, all you need to do is find the least common denominator. For 2 and 3, well, that's 6. Now in step 2, multiply every single term on both sides by that LCD of 6. And like magic, in step 3, the fractions are completely gone. We're just left with 3x plus 2x equals 60. Combine your like terms to get 5x equals 60, divide by 5, and boom, in step 4, x equals 12. No calculator, no messy fraction addition, just a clean whole number equation.

3:04 And guess what? The exact same trick works for decimals. You just multiply every term by a power of 10. Word problems and formulas. We're shifting our focus to translating plain English into math equations, which is an absolutely vital skill for the ASVAB. The crucial point here is just learning the vocabulary of algebra. Let's break down this phrase. 5 less than 3 times a number is 22. First, we assign a letter. Let's just use n to represent our unknown number. So 3 times a number becomes 3n. The word is translates directly to an equal sign. Now here is the tricky part. 5 less than actually means we subtract 5 at the very end, kind of working in reverse order. So step 1 gives us our perfectly translated equation. 3n minus 5 equals 22. From there, we just balance the scale. Add 5 to get 3n equals 27 in step 2, and then divide by 3 to get n equals 9 in step 3. Easy, right? All right, let's roll up our sleeves for section 4.

4:04 Step by step worked examples. We're going to meticulously execute seven specific examples exactly as they appear on the test to really build up your confidence. Let's get into it. Here's worked example 1. Step 1. Negative 2x plus 9 equals 1. First up, we subtract 9 from both sides, which brings us to step 2. Negative 2x equals negative 8. Now whenever the coefficient is negative, you have to divide by that exact negative number. Sign included. So we divide both sides by negative 2. Remember, a negative divided by a negative is a positive. So step 3 gives us x equals 4. Let's check out worked example 2, which brings parentheses into the mix. Step 1. 6 times the quantity x minus 2 plus 4 equals 22. Our very first move is to distribute that six to everything inside those parentheses. That gives us step 2. 6x minus 12 plus 4 equals 22. Now before you start doing anything across the equals sign, clean up your own house first.

5:06 Combine your like terms on the left. Negative 12 plus 4 is negative 8, leading to step 3. 6x minus 8 equals 22. Add 8 to both sides to get 6x equals 30, divide by 6, and there's step 4. x equals 5. Worked example 3 reinforces that beautiful fraction clearing rule we talked about earlier. Step 1. x over 4 minus 2 equals x over 6. The lowest common denominator for 4 and 6 is 12. So step 2, we multiply every single term by 12. And please remember, even that constant 2 gets multiplied. This gives us step 3. 3x minus 24 equals 2x. Subtract 2x from both sides, and you get x minus 24 equals 0. Add 24, and you're done. Step 4. x equals 24. Now for worked example 4, where we're dealing with a fractional coefficient. Step 1. 2 fifths x plus 3 equals 11. First things first, keep the balance by subtracting 3 from both sides. That leaves us with step 2.

6:10 2 fifths x equals 8. Here is a super fast shortcut. To get rid of a fraction that's multiplied by a variable, just multiply both sides by its reciprocal. Basically flip the fraction upside down. So in step 3, we multiply both sides by 5 over 2. The fraction on the left turns into a 1, and on the right, 8 times 5 halves equals 20. Step 4. x equals 20. Worked example 5 highlights two very special but super important edge cases. Sometimes when you're solving, the variable terms just cancel out completely on both sides. If you're left with a totally false statement, like 2 equals 5, that is a contradiction. It simply means there is no solution. But if you're left with a universally true statement, like 4x minus 12 equals 4x minus 12, well, that's called an identity. It means there are infinitely many solutions. Literally any real number you plug in for x is going to work. Let's look at worked example 6, which is all about rearranging a literal formula. This looks tricky because it's almost entirely letters, but the secret is to just treat those other letters exactly like numbers.

7:12 Step 1 shows the formula for the area of a triangle, a equals 1 half b times h, and we want to solve for h. First, let's clear out that fraction by multiplying both sides by 2, giving us step 2. 2a equals bh. Now, since b is multiplying h, we undo that by dividing both sides by b. That cleanly leaves us with step 3, h equals 2a over b. Finally, worked example 7 ties a little geometry back into our algebra. Imagine an isosceles triangle with 2 equal sides of length x, and a third side that's 6 units longer. The perimeter, which is just the total distance around the outside, is 39. So step 1 is building that equation. x plus x plus the quantity x plus 6 equals 39. Combine those like terms on the left side for step 2. 3x plus 6 equals 39. Subtract 6 to get step 3, 3x equals 33, divide by 3, and step 4 reveals that x equals 11. Okay, section 5, costly mistakes, and review. You've seen the right way to do things, but let's lean in a bit here because we need to look closely at the specific traps that constantly drain points from test takers on the ASVAB.

8:20 Here is perhaps the single most common error of all. When distributing a number across parentheses, like 3 times the quantity x plus 4, a lot of people rush and only multiply the first term. They write down 3x plus 4. That is definitely wrong. You have to distribute that 3 to every single thing inside the parentheses. The right way is 3 times x plus 3 times 4, which gives you 3x plus 12. Don't leave part of your product behind. Think of this as your mental diagnostic checklist to avoid losing easy points. First, dropping negative signs when dividing. Always, always keep the sign attached to the coefficient. Second, applying operations to only one side. Remember our balance scale? What you do to one side, you absolutely must do to the other. And finally, missing the substitution check. Honestly, this one is mandatory. When you get an answer, take a second to plug it back into the original equation. If both sides equal the same number, you know with 100% certainty you got it right. It takes seconds and it is entirely foolproof.

9:23 We have reached the final section. Section 6, check yourself final practice. I want to invite you to test out this new calculator-free confidence with two actual practice questions, straight from our source material. So grab that pencil and scratch paper. Here's question 1. What is the solution of negative 4x minus 3 equals 21? Is it A, negative 4.5, B, negative 6, C, 6, or D, 4.5? I'll give you a second to pause the explainer if you want to work the problem out on your paper right now. All right, let's check it. The correct answer is B, negative 6. Let's walk through it together. Step 1 is our equation, negative 4x minus 3 equals 21. We undo the subtraction by adding 3 to both sides, which hands us step 2, negative 4x equals 24. Finally, we divide both sides by negative 4. Remember that rule we just talked about? Keep the negative sign. 24 divided by negative 4 gives us step 3, x equals negative 6.

10:25 Outstanding. Now for the final question, a classic word translation challenge. 7 more than twice a number is 31. What is the number? Is it A, 8.5, B, 19, C, 12, or D, 24? Take one last quick pause to translate this English phrase into an equation and solve it. And the correct answer is C, 12. Let's build the equation in step 1. Twice a number is 2n, and 7 more than means we add 7 at the end. Is means equals. So we get 2n plus 7 equals 31. Subtract 7 from both sides for step 2, getting 2n equals 24. Divide by 2, and step 3 gives us n equals 12. Just look at what you've accomplished here today. Step by step, maintaining balance, and clearing out obstacles like fractions and negative signs. Think about how empowering it is to walk into a testing room knowing you can completely deconstruct and beat a math test with nothing more than pure logic, careful steps, and a simple pencil. You've got the tools now. So how many more points are you going to rack up on the ASVAB using these exact methods?

11:32 Keep practicing, and thanks for joining me.