Problem 13 - Algebra 1: Concepts and Skills

Question

Solve the equation. Check your solution in the original equation. $$ 4=-\frac{2}{3} x $$

Step-by-Step Solution

Answer

The solution to the equation is \( c = 3 \)

Step 1: Simplify the Left Side of the Equation

Distribute the 7 inside the bracket: \( 7 * c - 7 * 7 + 4c = 7c - 49 + 4c = 11c - 49 \)

Step 2: Simplify the Right Side of the Equation

Distribute the -2 inside the bracket: \( -2 * c - 2 * 5 = -2c -10 \)

Step 3: Set the Simplified Left Side of the Equation Equal to the Simplified Right Side

Write the simplified equation as: \( 11c - 49 = -2c - 10 \)

Step 4: Get All the c Terms on One Side and the Numerical Terms on the Other

Add 2c to both sides and add 49 to both sides to get: \( 11c + 2c = 49 - 10, which simplifies to 13c = 39 \)

Step 5: Solve for c

To solve for c, divide both sides of the equation by 13: \( c = 39 / 13 = 3 \)

Step 6: Substitute the Solution into the Original Equation to Check It

Substitute c = 3 into the original equation: \( 7(3 - 7) + 4 * 3 = -2(3 + 5) \). Simplifying both sides gives \( -8 = -8 \), confirming that the solution is correct. Therefore, the solution to the equation is \( c = 3 \)

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra used to simplify expressions and solve equations. It involves multiplying a single term by each term inside a parenthesis. This property helps to remove parentheses and combine terms for easier computation.

In the given exercise, the distributive property is applied twice. Here's how it works:

• On the left side of the equation, the expression is \(7(c-7)\). Using distributive property: \(7 \times c - 7 \times 7\), which simplifies to \(7c - 49\).
• On the right side, with the expression \(-2(c+5)\), it becomes \(-2 \times c - 2 \times 5\), simplifying to \(-2c - 10\).

Both operations make it easier to work with the equation without the complexity of brackets. Remember, for the distributive property: \(a(b + c) = ab + ac\). Applying this correctly is crucial to manage and simplify algebraic equations.

Combining Like Terms

Combining like terms is the process of simplifying algebraic expressions by merging terms that have the same variable component. When terms are combined, it reduces clutter and makes equations more straightforward.

In this exercise:

• Once the distributive property is applied, the equation becomes \(7c + 4c - 49 = -2c - 10\).
• Here, \(7c\) and \(4c\) are like terms as they both contain the variable \(c\). Adding them gives \(11c\).
• Now, the equation can be expressed in a more simplified form: \(11c - 49 = -2c - 10\).

Combining like terms reduces an equation to its simplest form, making it much easier to solve. Remember, like terms have identical variable parts and can thus be summed up or subtracted.

Checking Solutions

Once a solution is found, it is vital to verify if it actually satisfies the original equation. This is known as checking solutions. It helps ensure the accuracy of your work and confirms that you haven't made any calculation errors.

In our equation \(7(c-7) + 4c = -2(c+5)\), we determined \(c = 3\). To check:

• Substitute \(c\) back into the original equation: \(7(3 - 7) + 4 \times 3\) on the left, and \(-2(3 + 5)\) on the right.
• Calculate the expressions: \(7(-4) + 12 = -28 + 12 = -16\), and \(-2 \times 8 = -16\).
• Both simplify to \(-16\) which confirms the solution \(c = 3\) is correct.

Always substitute your solution back into the initial equation to ensure its correctness. This process not only provides confidence in your answer but also enhances overall understanding of algebraic manipulations.