Question

Solve each equation. Provide reasons for each step. Check the answer. $$ \frac{3}{8} p-\frac{4}{9}=\frac{5}{8} p+7 $$

   Solve each equation. Provide reasons for each step. Check the answer.
$$
\frac{3}{8} p-\frac{4}{9}=\frac{5}{8} p+7
$$

Intermediate Algebra: Connecting Concepts through Applications

Intermediate Algebra: Connecting Concepts through Applications

Mark Clark; Cynthia… 2nd Edition

Chapter 1, Problem 64

Instant Answer

Solved by Expert Alison Rodriguez

02/25/2022

Step 1

This is the multiplication property of equality. $$ 72 \cdot \frac{3}{8} p - 72 \cdot \frac{4}{9} = 72 \cdot \frac{5}{8} p + 72 \cdot 7 $$ This simplifies to: $$ 27p - 32 = 45p + 504 $$ Show more…

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Solve each equation. Provide reasons for each step. Check the answer. $$ \frac{3}{8} p-\frac{4}{9}=\frac{5}{8} p+7 $$

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Key Concepts

Checking Solutions

The process of checking solutions involves substituting the obtained value back into the original equation to ensure that both sides are balanced. This step verifies the correctness of the solution and is an important practice in algebra to confirm that no mistakes were made during the manipulation of the equation.

Operations with Fractions

Operations with fractions are crucial when working with equations containing fractional coefficients. Techniques such as finding a common denominator, multiplying both sides of an equation by the least common multiple, and simplifying fractions are used to manage and eliminate fractions, simplifying the equation and making it easier to solve.

Properties of Equality

The properties of equality govern how operations can be performed on both sides of an equation while maintaining the balance between them. These include adding, subtracting, multiplying, or dividing both sides by the same quantity. Mastery of these properties is essential for transforming and simplifying equations correctly during the solving process.

Linear Equations

Linear equations are mathematical statements that involve a variable raised only to the first power, combined with constant terms. The objective is to isolate the variable on one side of the equation, often by combining like terms and using inverse operations. This concept is fundamental in algebra and underpins the techniques used to solve equations systematically.

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Transcript

00:01 In this problem, we're going to be solving the equation that's on the screen.

00:04 And as we go, we're going to justify or provide this each reason for each step.

00:08 So in here, i'm going to first start by getting rid of the denominators.

00:13 So i'm going to multiply through by the common denominator.

00:16 In this case, it's the number 72.

00:18 So because we're multiplying something on both sides of the equation, this is the multiplication property of equality.

00:25 Now, when i do multiplication property of equality, i'm going to put p -o -e, and that will allow us to reduce the writing for poe.

00:34 So then from there, look ahead of multiply.

00:38 So 72 times three eighths.

00:41 So 72 divided by eight is nine, and nine times three is 27.

00:45 So i have 27 p.

00:47 Now over for the next one one i'd have distributed through.

00:49 So 72 divided by nine is eight times four is 32.

00:54 Again, 72 divided by eight is nine, nine times five is 45.

00:58 And then the only other one here is seven, and that one we can't cancel anything out.

01:01 So 7 times 72 is 504.

01:04 So what i just said here was distributive property.

01:07 So this is the distributive property of equality.

01:10 So again, p -o -e, just to shorten things down.

01:13 All right.

01:13 So now from there, i'm going to kind of move things around.

01:16 So to get variables on the left -hand side, so to isolate that variable, i'm going to subtract 45p from both sides.

01:23 So because i'm subtracting, this is our subtraction property of equality.

01:28 Okay.

01:29 So when i actually do the subtraction, 27 minus 45 is negative 18p.

01:36 Now from there, i'm going to go ahead and move my 32 over to the right hand side.

01:41 So i'm going to add 32 and add 32.

01:46 And i'm going to go ahead and write down negative 18p.

01:49 And then i'm going to add 504 plus 32.

01:52 And that gets me a total of 536.

01:55 Because i added on both sides, this is my addition property of equality.

02:01 So a dpov.

02:02 All right...

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