Solve 7(1+4p)=1 For P

by Alex Johnson 22 views

Understanding the Equation

We are tasked with simplifying the expression 7(1+4p)=1 and solving for the variable p. This is a common type of algebraic equation that involves distribution and isolating the variable. The goal is to find the value of p that makes the equation true. This process is fundamental in algebra and helps build a strong foundation for more complex mathematical problems. When we simplify an expression, we aim to rewrite it in its most basic form, often by combining like terms or removing parentheses. In this specific case, we have a number multiplied by a binomial inside parentheses, and this entire expression is set equal to a constant. This structure immediately suggests that the distributive property will be our first step in unraveling the equation and making it easier to work with.

Applying the Distributive Property

The first step in solving 7(1+4p)=1 is to apply the distributive property. This property states that for any numbers a, b, and c, the equation a(b+c) = ab + ac holds true. In our equation, a is 7, b is 1, and c is 4p. So, we multiply 7 by each term inside the parentheses:

  • 7 * 1 = 7
  • 7 * 4p = 28p

After applying the distributive property, the equation becomes:

7 + 28p = 1

This step is crucial because it removes the parentheses, transforming the equation into a simpler linear form. It's a fundamental algebraic manipulation that allows us to proceed with isolating the variable. Many students find this step straightforward, but it's important to be meticulous, ensuring that each term within the parentheses is multiplied by the factor outside. A common error here is to forget to multiply the second term (4p) by 7, or to make a mistake in the multiplication itself. Double-checking this step can save a lot of trouble later on. The visual representation of this step is expanding the grouped terms into separate, additive terms, making the structure of the equation more apparent and manageable for subsequent operations.

Isolating the Variable Term

Now that we have the equation 7 + 28p = 1, our next goal is to isolate the term containing the variable, which is 28p. To do this, we need to eliminate the constant term (7) from the left side of the equation. We can achieve this by subtracting 7 from both sides of the equation, maintaining the balance of the equality:

7 + 28p - 7 = 1 - 7

This simplifies to:

28p = -6

This step is all about getting the variable term by itself on one side of the equation. Think of it like tidying up a desk; you move all the papers (variable terms) to one side and all the pens (constants) to the other. Subtracting 7 from both sides is the inverse operation of the addition that was present. By performing the same operation on both sides, we ensure that the equation remains true. This isolation of the variable term is a pivotal moment in solving the equation, as it brings us one step closer to finding the specific value of p. It’s essential to perform the subtraction accurately, paying close attention to the signs of the numbers involved. A miscalculation here, such as incorrectly subtracting 7 from 1, can lead to an incorrect final answer. This methodical approach of undoing operations step-by-step is the core of solving algebraic equations.

Solving for the Variable

With the equation now in the form 28p = -6, the final step to solve for p is to undo the multiplication of 28 with p. The inverse operation of multiplication is division. Therefore, we will divide both sides of the equation by 28:

28p / 28 = -6 / 28

This gives us:

p = -6 / 28

This fraction can be simplified. Both 6 and 28 are divisible by 2. So, we divide the numerator and the denominator by their greatest common divisor, which is 2:

  • -6 ÷ 2 = -3
  • 28 ÷ 2 = 14

Therefore, the simplified value of p is:

p = -3/14

This final step is where we determine the exact numerical value of our variable. Dividing both sides by the coefficient of p (which is 28) isolates p completely. Simplifying the resulting fraction is an important part of presenting the answer in its neatest form. It’s always good practice to reduce fractions to their lowest terms, as it makes the answer easier to understand and compare. This simplification process demonstrates a complete understanding of fractions and their properties. In essence, we've taken an initial equation that looked a bit complex and, through a series of logical and consistent algebraic steps, reduced it to a clear statement of the value of p. This journey from a complex expression to a simple solution is the essence of algebraic problem-solving.

Verification

To ensure our solution is correct, we can substitute p = -3/14 back into the original equation 7(1+4p)=1:

  • 7(1 + 4(-3/14))*
  • 7(1 - 12/14)
  • 7(1 - 6/7) (Simplifying the fraction 12/14 by dividing both numerator and denominator by 2)
  • 7(7/7 - 6/7) (Finding a common denominator for 1 and 6/7)
  • 7(1/7)
  • 1

Since the left side of the equation equals the right side (1 = 1), our solution p = -3/14 is correct. This verification step is a highly recommended practice in mathematics to confirm the accuracy of your calculations. It involves plugging the value you found for the variable back into the original equation and checking if the equality holds true. This provides confidence in your answer and helps catch any potential errors made during the solving process. It’s like a final check before submitting a piece of work; it ensures everything is in order. For more on solving linear equations, you can visit Khan Academy.