Math Expressions: Solve For P, Q, And R

Hey guys! Ever feel like math problems are just a bunch of letters and numbers throwing a party without you? Well, today we're crashing that party and showing you how to make sense of it all. We're diving into evaluating expressions, which is basically just plugging in values for variables and doing the math. It's a super handy skill, not just for your math class, but for understanding how formulas work in the real world. We've got a bunch of expressions here, and we're going to solve them step-by-step with $p=-4$, $q=8$, and $r=-10$. Get ready to flex those math muscles!

Understanding the Basics: Substitution and Order of Operations

Before we jump into the nitty-gritty, let's quickly go over what we're doing. Evaluating an expression means replacing the variables (those letters like $p$, $q$, and $r$) with their given numerical values. Once we've done that, we follow the order of operations, often remembered by the acronym PEMDAS or BODMAS. This handy little guide tells us the correct sequence to perform calculations: Parentheses (or Brackets), Exponents (or Orders), Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Sticking to this order is crucial for getting the right answer. Mess it up, and you'll end up with a totally different, and usually incorrect, result. So, keep PEMDAS in mind as we work through these problems, because it's our best friend in making sure our calculations are accurate and efficient.

Expression 1: $-p^2-7q+r^2$

Alright, let's kick things off with our first expression: $-p^2-7q+r^2$. We need to substitute $p=-4$, $q=8$, and $r=-10$ into this bad boy. Remember, when you substitute a negative number, especially when it's being squared or multiplied, it's a good idea to use parentheses to avoid confusion. So, $-p^2$ becomes $-(-4)^2$. This is a common sticking point for a lot of people, so pay attention! The exponent (the '2') applies only to the number inside the parentheses, which is -4. Squaring -4 means multiplying it by itself: $(-4) imes (-4) = 16$. So, $-(-4)^2$ is actually $-16$, not 16. Always be careful with those negative signs and exponents, guys!

Now, let's plug in the rest of the values. $-7q$ becomes $-7 imes 8$, which is $-56$. And $r^2$ becomes $(-10)^2$. Again, using parentheses helps us remember that we're squaring -10. $(-10) imes (-10) = 100$. So, our expression now looks like: $-16 - 56 + 100$.

We're left with only addition and subtraction, so we work from left to right. First, $-16 - 56$. Subtracting a larger number from a smaller one results in a negative number. Think of it like starting at -16 on a number line and moving 56 steps to the left. That takes us to $-72$. So now we have $-72 + 100$.

Finally, $-72 + 100$. We're adding a positive number to a negative number. This is the same as $100 - 72$. Imagine you have $100 and you owe $72. You'll have $28 left. So, the value of this expression is 28. Pretty straightforward once you break it down, right? Keep that PEMDAS order locked in!

Expression 2: $-p^2-2q+r$

Moving on to our second expression, we have $-p^2-2q+r$. This one looks pretty similar to the first, but there's a subtle difference that can change the outcome if we're not careful. We'll use the same values: $p=-4$, $q=8$, and $r=-10$. Let's substitute them in, remembering those parentheses for the negative numbers.

$-p^2$ once again becomes $-(-4)^2$. As we established, the exponent applies to the -4, so $(-4)^2 = 16$. Therefore, $-p^2 = -16$. It's super important to get this part right, as it's a common trap. The negative sign outside the parentheses is not part of the squaring operation itself.

Next, $-2q$ becomes $-2 imes 8$, which equals $-16$. And finally, $+r$ is simply $+(-10)$, which is $-10$.

So, our expression transforms into $-16 - 16 - 10$. Now, we just work from left to right, performing the subtractions.

First, $-16 - 16$. When you subtract 16 from -16, you move further into the negative numbers. This equals $-32$. So, we now have $-32 - 10$.

Finally, $-32 - 10$. Subtracting 10 from -32 takes us even further down the number line, resulting in $-42$. So, the value of the expression $-p^2-2q+r$ with the given values is -42. See how a small change in the expression can lead to a completely different answer? That's why paying attention to every single detail is key in math!

Expression 3: rac{q+r}{q+p}

Alright, mathletes, let's tackle some fractions! Our third expression is rac{q+r}{q+p}. This involves addition in both the numerator (the top part) and the denominator (the bottom part), and we need to be careful about the order of operations here. Remember, anything inside parentheses (or implied parentheses, like the division bar separating the top and bottom) needs to be calculated before the division occurs. We're still using $p=-4$, $q=8$, and $r=-10$.

Let's start with the numerator: $q+r$. Substituting the values, we get $8 + (-10)$. Adding a negative number is the same as subtracting its positive counterpart. So, $8 - 10 = -2$. The numerator is -2.

Now, let's work on the denominator: $q+p$. Plugging in the numbers, we have $8 + (-4)$. Again, adding a negative is like subtracting. So, $8 - 4 = 4$. The denominator is 4.

Our expression now looks like rac{-2}{4}. We need to simplify this fraction. Both the numerator and the denominator are divisible by 2. So, we divide both by 2: $-2 ilde{/} 2 = -1$ and $4 ilde{/} 2 = 2$. This gives us the simplified fraction rac{-1}{2}.

So, the value of rac{q+r}{q+p} is -rac{1}{2} or -0.5. Remember to always simplify your fractions if possible! It makes the answer cleaner and easier to understand. Good job working through that fraction, guys!

Expression 4: rac{p+r}{p+q}

We're on a roll, and this next expression, rac{p+r}{p+q}, is another fraction. It's very similar to the last one, just with the variables switched around a bit in the denominator. Let's substitute our trusty values: $p=-4$, $q=8$, and $r=-10$.

First, let's calculate the numerator: $p+r$. This becomes $(-4) + (-10)$. Adding two negative numbers means we combine their magnitudes and keep the negative sign. So, $-4 - 10 = -14$. The numerator is -14.

Next, let's calculate the denominator: $p+q$. Substituting the values, we get $(-4) + 8$. Adding a negative number to a positive number means we find the difference between their absolute values and use the sign of the number with the larger absolute value. The difference between 8 and 4 is 4, and 8 has the larger absolute value (and is positive), so the result is 4.

Our fraction is now rac{-14}{4}. Like the last fraction, this can be simplified. Both -14 and 4 are divisible by 2. Dividing the numerator by 2 gives us -7. Dividing the denominator by 2 gives us 2. So, the simplified fraction is rac{-7}{2}.

Thus, the value of rac{p+r}{p+q} is -rac{7}{2} or -3.5. Nicely done! Fractions can be tricky, but by breaking them down and simplifying, we conquer them.

Expression 5: rac{3 q}{r}-rac{5}{p}

Time for some more complex fractions with subtraction involved! Our fifth expression is rac{3 q}{r}-rac{5}{p}. This involves multiplication within the fractions and then subtraction between them. We'll use $p=-4$, $q=8$, and $r=-10$.

Let's tackle the first fraction: rac{3q}{r}. First, calculate the numerator: $3q = 3 imes 8 = 24$. So, the first fraction is rac{24}{-10}. We can simplify this fraction by dividing both numerator and denominator by 2, giving us rac{12}{-5}, or -rac{12}{5}.

Now for the second fraction: rac{5}{p}. This becomes rac{5}{-4}, or -rac{5}{4}.

Our expression now is -rac{12}{5} - (-rac{5}{4}). Subtracting a negative is the same as adding a positive. So, this becomes -rac{12}{5} + rac{5}{4}.

To add these fractions, we need a common denominator. The least common multiple of 5 and 4 is 20. To convert -rac{12}{5} to have a denominator of 20, we multiply both the numerator and denominator by 4: (-rac{12 imes 4}{5 imes 4}) = -rac{48}{20}.

To convert rac{5}{4} to have a denominator of 20, we multiply both the numerator and denominator by 5: (rac{5 imes 5}{4 imes 5}) = rac{25}{20}.

Now we can add them: -rac{48}{20} + rac{25}{20} = rac{-48 + 25}{20}.

$-48 + 25 = -23$. So, the result is rac{-23}{20}.

Therefore, the value of rac{3 q}{r}-rac{5}{p} is -rac{23}{20} or -1.15. Working with fractions requires a good understanding of common denominators and keeping track of signs!

Expression 6: rac{3 r}{q}-rac{2}{r}

We're getting closer to the end, guys! Our sixth expression is rac{3 r}{q}-rac{2}{r}. This also involves fractions and subtraction. Let's plug in $p=-4$, $q=8$, and $r=-10$.

First fraction: rac{3r}{q}. The numerator is $3r = 3 imes (-10) = -30$. So, the fraction is rac{-30}{8}. We can simplify this by dividing both by 2, giving us rac{-15}{4}.

Second fraction: rac{2}{r}. This becomes rac{2}{-10}. Simplifying this by dividing both by 2 gives us rac{1}{-5}, or -rac{1}{5}.

Our expression is now rac{-15}{4} - (-rac{1}{5}). Subtracting a negative is adding a positive, so we have rac{-15}{4} + rac{1}{5}.

To add these, we need a common denominator. The least common multiple of 4 and 5 is 20. Convert rac{-15}{4} by multiplying numerator and denominator by 5: (rac{-15 imes 5}{4 imes 5}) = -rac{75}{20}.

Convert rac{1}{5} by multiplying numerator and denominator by 4: (rac{1 imes 4}{5 imes 4}) = rac{4}{20}.

Now add them: -rac{75}{20} + rac{4}{20} = rac{-75 + 4}{20}.

$-75 + 4 = -71$. So, the result is rac{-71}{20}.

Thus, the value of rac{3 r}{q}-rac{2}{r} is -rac{71}{20} or -3.55. Keep practicing those fraction skills!

Expression 7: rac{5 r}{2 p-3 r}

Alright, this next one, rac{5 r}{2 p-3 r}, looks a bit more intimidating because of the operations within the denominator. But don't sweat it, guys! We just need to be extra careful with the order of operations. We have $p=-4$, $q=8$, and $r=-10$.

Let's start with the numerator: $5r$. This is $5 imes (-10) = -50$. The numerator is -50.

Now, let's focus on the denominator: $2p-3r$. This is where we need to follow PEMDAS carefully. First, perform the multiplications:

$2p = 2 imes (-4) = -8$.

$3r = 3 imes (-10) = -30$.

So, the denominator becomes $-8 - (-30)$. Subtracting a negative number is the same as adding its positive counterpart. Thus, $-8 - (-30)$ becomes $-8 + 30$.

Calculating $-8 + 30$: The difference between 30 and 8 is 22, and since 30 is the larger number (and positive), the result is 22. The denominator is 22.

Our expression is now rac{-50}{22}. We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2.

$-50 ilde{/} 2 = -25$.

$22 ilde{/} 2 = 11$.

So, the simplified fraction is rac{-25}{11}.

Therefore, the value of rac{5 r}{2 p-3 r} is -rac{25}{11} or approximately -2.27. You're crushing it!

Expression 8: rac{3 q}{3 p-2 q}

Last one, folks! Let's conquer rac{3 q}{3 p-2 q} with $p=-4$, $q=8$, and $r=-10$. This expression also has operations in the denominator, so precision is key.

First, let's find the numerator: $3q$. This is $3 imes 8 = 24$. The numerator is 24.

Now, the denominator: $3p-2q$. We need to perform the multiplications first:

$3p = 3 imes (-4) = -12$.

$2q = 2 imes 8 = 16$.

So, the denominator becomes $-12 - 16$.

Calculating $-12 - 16$: When you subtract 16 from -12, you move further into the negative numbers. This equals $-28$. The denominator is -28.

Our expression is now rac{24}{-28}. We can simplify this fraction. Both 24 and -28 are divisible by 4.

$24 ilde{/} 4 = 6$.

$-28 ilde{/} 4 = -7$.

So, the simplified fraction is rac{6}{-7}, which is the same as -rac{6}{7}.

And there you have it! The value of rac{3 q}{3 p-2 q} is -rac{6}{7} or approximately -0.86. We've successfully evaluated all the expressions!

Wrapping It Up

Phew! We made it through all eight expressions. Evaluating expressions might seem like a basic skill, but it's the foundation for so much more in mathematics. Remember to always pay attention to the signs, especially with negative numbers and exponents. And never forget the order of operations (PEMDAS/BODMAS) – it's your roadmap to getting the correct answer every time. Keep practicing these types of problems, guys, and you'll become math wizards in no time. If you ever get stuck, just break the problem down, step by step. You've got this!