Problem 11 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve equation. Be sure to check your proposed solution by substituting it for the variable in the original equation. \(38=30-2(x-1)\)

Step-by-Step Solution

Answer

The solution to the equation is \(x=-3\).

Step 1: Distribute

First, distribute -2 to the terms inside the parenthesis. This will result in the equation: \(38=30-2x+2\).

Step 2: Simplify

Combine the constant terms on the right side of the equation: \(38=32-2x\). Then, get all constant terms on one side by subtracting 32 from both sides, yielding: \(6=-2x\).

Step 3: Solve for x

Now, isolate the variable \(x\) by dividing every term by -2. This produces the solution: \(x=-3\).

Step 4: Check the Solution

Finally, substitute -3 for \(x\) in the original equation to confirm that the left-hand side equals the right-hand side. Substitution results in: \(38=30-2(-3-1)\). Simplifying, we find: \(38=38\), confirming that -3 is a solution to the equation.

Key Concepts

Equation Solving

Equation solving is a fundamental aspect of algebra. When we solve an equation, we're looking for the values of variables that make the equation true. In the given problem, we begin with the equation \[ 38 = 30 - 2(x - 1) \].Solving an equation generally involves systematically isolating the variable. This means performing operations that simplify the equation and bring the variable to one side. For the present problem, it involves distributing, simplifying, and then isolating the variable. The primary goal in equation solving is to manipulate the equation in a way that uncovers the value of the unknown variable. Ensure to perform the same operation on both sides of the equation to maintain balance. This ensures that the equation holds true throughout your calculations.

Distributive Property

The distributive property is a key tool in algebra, especially for expanding expressions and simplifying them. It states that \[ a(b + c) = ab + ac \]. This means you multiply each term inside the parenthesis by the term outside.In our example equation \[ 38 = 30 - 2(x - 1) \], the distributive property is used to remove the parenthesis: \[ -2(x - 1) \] becomes \[ -2x + 2 \], since \( -2 \) multiplied by \( x \) gives \( -2x \) and \( -2 \) multiplied by \( -1 \) results in \( +2 \).Using the distributive property effectively simplifies equations, making them easier to solve. It's important to apply it carefully, ensuring each term within the parenthesis is considered.

Checking Solutions

In mathematics, checking your solutions is crucial. This verifies that your calculated value satisfies the original equation. For the equation \[ 38 = 30 - 2(x - 1) \], after finding \( x = -3 \), you should substitute \( -3 \) back into the original equation to ensure correctness.Perform substitution by inserting \( x = -3 \) where \( x \) appears in the equation:\[ 38 = 30 - 2(-3 - 1) \].Simplify to verify:

• Calculate inside the parenthesis: \(-3 - 1 = -4 \)
• Next, multiply: \(-2 \times -4 = 8 \)
• Add to check: \(30 + 8 = 38 \)

Both sides match, confirming \( x = -3 \) is indeed the solution. Always perform such checks to avoid errors and to ensure that your solution is correct.