Solving Basic Algebraic Equations

Hey guys! Let's dive into solving some basic algebraic equations. We've got four problems here that will help us brush up on our algebra skills. We'll take it step by step to make sure everyone's on the same page. Let's get started!

a) ${25z + 49 = 149}$

Alright, our first equation is ${25z + 49 = 149}$. The goal here is to isolate ${z}$ on one side of the equation. To do that, we'll first subtract 49 from both sides:

${25z + 49 - 49 = 149 - 49}$

This simplifies to:

${25z = 100}$

Now, to completely isolate ${z}$, we need to divide both sides by 25:

${\frac{25z}{25} = \frac{100}{25}}$

Which gives us:

${z = 4}$

So, the solution to the first equation is ${z = 4}$. We can check our work by plugging the value of ${z}$ back into the original equation:

${25(4) + 49 = 100 + 49 = 149}$

Yep, it checks out! This confirms that our solution is correct. Remember, the key is to perform the same operation on both sides of the equation to maintain balance and eventually isolate the variable we're solving for. Understanding this basic principle makes solving linear equations much easier. It's all about keeping things balanced and moving step by step. With practice, you'll be solving these equations in your sleep! Keep an eye on those signs (positive and negative) and make sure you're applying the correct operations. Algebra can be fun once you get the hang of it. Also, always double-check your work by substituting the solution back into the original equation. This simple check can save you from making silly mistakes.

b) ${13 + 10t = 163}$

Next up, we have the equation ${13 + 10t = 163}$. Our mission is to isolate ${t}$. First, we'll subtract 13 from both sides of the equation:

${13 + 10t - 13 = 163 - 13}$

This simplifies to:

${10t = 150}$

Now, we divide both sides by 10 to get ${t}$ by itself:

${\frac{10t}{10} = \frac{150}{10}}$

Which results in:

${t = 15}$

So, ${t = 15}$ is the solution to the second equation. Let's verify by substituting ${t}$ back into the original equation:

${13 + 10(15) = 13 + 150 = 163}$

It's correct! Solving for ${t}$ involved subtracting the constant term and then dividing by the coefficient of ${t}$. This is a common pattern in solving linear equations. Recognizing these patterns can make solving equations faster and more efficient. Always remember to keep the equation balanced by performing the same operations on both sides. This ensures that the equality remains true. Also, be careful with the order of operations. In this case, we subtracted before dividing, but depending on the equation, the order might be different. Understanding the order of operations (PEMDAS/BODMAS) is crucial in algebra. And don't forget to check your solutions! It's a simple step that can save you a lot of headaches.

c) ${9y - 54 = 162}$

Moving on, we have the equation ${9y - 54 = 162}$. We want to isolate ${y}$. Start by adding 54 to both sides:

${9y - 54 + 54 = 162 + 54}$

This simplifies to:

${9y = 216}$

Now, divide both sides by 9:

${\frac{9y}{9} = \frac{216}{9}}$

Which gives us:

${y = 24}$

Therefore, the solution to the third equation is ${y = 24}$. Let’s plug it back into the original equation to check:

${9(24) - 54 = 216 - 54 = 162}$

Spot on! To solve this one, we added the constant term to both sides and then divided by the coefficient of ${y}$. The steps are pretty straightforward once you get the hang of it. Always double-check your arithmetic to avoid mistakes. Simple errors in addition, subtraction, multiplication, or division can throw off the entire solution. Also, remember that practice makes perfect. The more equations you solve, the more comfortable you'll become with the process. Try solving similar equations on your own to reinforce your understanding. And if you ever get stuck, don't hesitate to ask for help. There are plenty of resources available online and in textbooks. Keep practicing, and you'll become an algebra whiz in no time!

d) ${181 - 8r = 45}$

Finally, we have the equation ${181 - 8r = 45}$. We need to isolate ${r}$. First, subtract 181 from both sides:

${181 - 8r - 181 = 45 - 181}$

This simplifies to:

${-8r = -136}$

Now, divide both sides by -8:

${\frac{-8r}{-8} = \frac{-136}{-8}}$

Which gives us:

${r = 17}$

So, the solution to the last equation is ${r = 17}$. Let's check our answer:

${181 - 8(17) = 181 - 136 = 45}$

It works! Notice that we divided by a negative number in this case. Remember to pay close attention to the signs when working with negative numbers. A common mistake is to forget the negative sign, which can lead to an incorrect solution. Also, be aware of the order of operations. In this equation, we subtracted before dividing. And as always, double-check your work to catch any errors. Solving equations like this involves a few steps, but with practice, you'll be able to solve them quickly and accurately. Keep practicing, and you'll become more confident in your algebra skills. Understanding how to manipulate equations and isolate variables is a fundamental skill in mathematics, and it will be useful in many areas of study and in real-life situations. So, keep up the great work!

We've successfully solved all four equations! I hope this was helpful. Keep practicing, and you'll become an algebra pro in no time. Remember to always double-check your answers and take it one step at a time. You got this!