Problem 36 - Algebra 2

Question

Solve each equation. Check your solution. $$ -7(p+7)+3(p-4)=-17 $$

Step-by-Step Solution

Answer

The solution is \(p = -11\).

Step 1: Distribute the terms

First, distribute the multiplication across the expression within the parentheses. For the equation \(-7(p+7)+3(p-4)=-17\), distribute \(-7\) to each term in \((p+7)\) and \(3\) to each term in \((p-4)\):\[-7(p+7) = -7p - 49\]\[3(p-4) = 3p - 12\]

Step 2: Combine like terms

Combine the terms from the distribution:\[-7p - 49 + 3p - 12 = -17\]Combine the like terms:\[(-7p + 3p) = -4p\]\[(-49 - 12) = -61\]So, the equation becomes:\[-4p - 61 = -17\]

Step 3: Isolate the variable term

Add 61 to both sides of the equation to move the constant term to the right side:\[-4p - 61 + 61 = -17 + 61\]This simplifies to:\[-4p = 44\]

Step 4: Solve for the variable

Divide both sides by \(-4\) to solve for \(p\):\[p = \frac{44}{-4}\]Simplifying this gives:\[p = -11\]

Step 5: Check the solution

Substitute \(p = -11\) back into the original equation to verify the solution:\[-7((-11)+7) + 3((-11)-4) = -17\]Calculate:\[-7(-4) = 28\]\[3(-15) = -45\]Add the products:\[28 - 45 = -17\]The left side equals the right side (-17), confirming that the solution \(p = -11\) is correct.

Key Concepts

Distributive Property

The distributive property is a fundamental property of algebra. It allows you to simplify expressions by multiplying a single term across terms within parentheses. In the equation \[-7(p+7) + 3(p-4) = -17\]we apply the distributive property as follows:- Multiply \(-7\) with each term inside \((p+7)\). This gives us: \(-7p - 49\).- Similarly, multiply \(3\) with each term inside \((p-4)\). This results in: \(3p - 12\).By applying the distributive property, we transform complex expressions into simpler ones, making it easier to solve equations.

Combining Like Terms

Combining like terms is a technique used to simplify expressions. It involves adding or subtracting terms that have identical variable parts. For the equation\[-7p - 49 + 3p - 12 = -17\]we identify and combine like terms:- Combine the \(p\) terms: \(-7p + 3p = -4p\).- Combine the constant terms: \(-49 - 12 = -61\).The equation then simplifies to \(-4p - 61 = -17\). This step reduces the equation to fewer terms, making it easier to focus on solving for the variable.

Variable Isolation

Variable isolation is a strategy used to solve for the unknown in an equation. The goal is to manipulate the equation so that the variable is by itself on one side of the equation. Starting with \[-4p - 61 = -17\]we isolate \(p\) by performing the following steps:

• Add \(61\) to both sides to eliminate the constant term: \[-4p - 61 + 61 = -17 + 61\], simplifying to \[-4p = 44\].
• Divide by \(-4\) to solve for \(p\): \[p = \frac{44}{-4}\], which simplifies to \(p = -11\).

Isolating the variable allows you to find the solution in a clear, straightforward manner.

Solution Verification

Solution verification is an important last step in solving equations. It involves substituting the found solution back into the original equation to ensure that it satisfies the equation.Let's verify our solution \(p = -11\) in the original equation:\[-7((-11) + 7) + 3((-11) - 4) = -17\]

• Calculate each term: \[-7(-4) = 28\] and \[3(-15) = -45\].
• Add these results: \[28 - 45 = -17\].

The left side equals the right side \(-17\), confirming that our solution \(p = -11\) is correct. Verification ensures that our solution not only satisfies the equation but also is mathematically accurate.