Problem 9 - College Algebra Essentials

Question

Solve and check linear equation. \(3(x-2)+7=2(x+5)\)

Step-by-Step Solution

Answer

The solution to the equation is x = 9

Step 1: Expand the Brackets

Start by removing brackets in the equation. To do this, use the distributive property to simplify the equation: \(3(x - 2) = 3x - 6\) and \(2(x + 5) = 2x + 10\). The equation then becomes: \(3x - 6 + 7 = 2x + 10\)

Step 2: Simplify

Next, combine like terms on both sides of the equation to simplify it. On the left side, combine -6 and +7 to get +1, and, on the right side, the equation remains the same since there are no like terms. The simplified equation therefore is: \(3x + 1 = 2x + 10\)

Step 3: Rearrange the Equation

Next, get all terms containing the variable 'x' on the left side of the equation and the constants on the right side by subtracting '2x' from both sides, and subtracting '1' from both sides. This results in: \(3x - 2x = 10 - 1\) which simplifies to \(x = 9\)

Step 4: Check the Solution

Substitute x = 9 back into the original equation to confirm if it's indeed the solution. The original equation is: \(3(x-2)+7=2(x+5)\). Substituting x = 9 gives: \(3(9-2)+7=2(9+5)\) which simplifies to \(21=28\). Since this is not true, there was a mistake in the calculation. Let's find the correct answer.

Step 5: Correct Calculation

Combining like terms in step 2 resulted in a mistake. Correctly, -6 and +7 result in +1. Therefore, the equation after simplifying is: \(3x + 1 = 2x + 10\). Getting all terms with x to one side gives \(3x - 2x = 10 - 1\) resulting in \(x = 9\).

Step 6: Correctly Check the Solution

Substitute x = 9 back into the original equation to confirm if it's indeed the solution. The original equation is: \(3(x - 2) + 7 = 2(x + 5)\). Substituting x = 9 gives: \(3(9-2) + 7 = 2(9 + 5)\), which simplifies to \(28 = 28\), confirming that 'x = 9' is the correct solution.

Key Concepts

Distributive Property

The distributive property is a key principle in algebra that helps simplify expressions and equations. It states that if you have a multiplication operation outside a parenthesis, you need to multiply each term inside the parenthesis separately. This principle is written mathematically as:

• \( a(b + c) = ab + ac \).

Let's consider the equation from the exercise: \( 3(x-2)+7=2(x+5) \). To use the distributive property, multiply the numbers outside the parenthesis with each term inside the parenthesis:

• \( 3(x-2) = 3x - 6 \)
• \( 2(x+5) = 2x + 10 \)

Now the equation looks cleaner: \( 3x - 6 + 7 = 2x + 10 \). Simplifying in this way turns a complex equation into something more manageable and sets the stage for solving it.

Solving Equations

Solving equations is all about finding the value of the unknown variable that makes the equation true. In our case, we're dealing with a linear equation, meaning an equation where the highest power of the variable is 1.

• Start by simplifying both sides of the equation. Combine like terms to make the equation look cleaner.
• After using the distributive property on \( 3(x - 2) \) and \( 2(x + 5) \), our equation is \( 3x - 6 + 7 = 2x + 10 \).
• Simplify further by combining \(-6 + 7\) to get \(+1\). The equation becomes \( 3x + 1 = 2x + 10 \).
• Next, get all the terms with 'x' on one side. Subtract \(2x\) from both sides:
  \( 3x - 2x + 1 = 2x - 2x + 10 \). This simplifies to \( x + 1 = 10 \).
• Remove constants from the side with the variable by subtracting \(1\) from both sides:
  \( x + 1 - 1 = 10 - 1 \). This simplifies to \( x = 9 \).

By isolating \( x \), you have solved the equation, determining that \( x \) equals 9.

Checking Solutions

Once you've solved the equation, it's crucial to verify your solution. This step ensures the integrity and correctness of your answer. To check the solution \( x = 9 \), substitute it back into the original equation and verify both sides:

1. Start with replacing \( x \) by 9 in the equation:
   \( 3(x-2) + 7 = 2(x+5) \) becomes \( 3(9-2) + 7 = 2(9+5) \).
2. Simplify both sides separately:
   • The left side: \( 3(7) + 7 = 21 + 7 = 28 \).
   • The right side: \( 2(14) = 28 \).
3. Check if both sides are equal; here, \( 28 = 28 \) confirms they match.

If the equation holds true, as it does here, then \( x = 9 \) is indeed the correct solution. This provides assurance that you've worked through the problem correctly and found the accurate answer.