Problem 53 - Elementary Algebra

Question

Solve. $$ 6(x-2)-(7 x-12)=14 $$

Step-by-Step Solution

Answer

The solution is \(x = -14\).

Step 1: Expand the Equation

We start with the equation: \[6(x-2)-(7x-12)=14\]First, we distribute the numbers outside the parentheses into the terms inside.Expanding the first term: \[6(x-2) = 6x - 12\]Expanding the second term: \[-(7x - 12) = -7x + 12\]After distributing, our equation becomes:\[6x - 12 - 7x + 12 = 14\]

Step 2: Combine Like Terms

Next, we combine the terms with \(x\) and the constant terms:\[(6x - 7x) + (-12 + 12) = 14\]Simplify to get:\[-x = 14\]

Step 3: Solve for x

Now, we solve for \(x\). We have:\[-x = 14\]To solve for \(x\), multiply both sides by -1:\[x = -14\]

Key Concepts

Distributive Property

The distributive property allows us to multiply a number by a group of numbers added or subtracted together. When it's written out, it looks like this: \(a(b + c) = ab + ac\). It is most useful when dealing with expressions inside parentheses. In the equation given, \(6(x-2)\), the "6" is distributed, or multiplied, by both \(x\) and \(-2\). This means:

• \(6 \times x = 6x\)
• \(6 \times -2 = -12\)

Thus, \(6(x-2)\) becomes \(6x - 12\). The same principle applies when we encounter a negative sign before a set of parentheses, as in \( -(7x - 12) \), which when expanded becomes \(-7x + 12\). Remember, distributing a negative sign changes the signs of all terms within the parentheses.

Combining Like Terms

Combining like terms is the process of simplifying expressions by adding or subtracting terms that have the same variable raised to the same power. In our expanded equation, \(6x - 12 - 7x + 12 = 14\), we have two types of terms to combine:

• Variable terms: \(6x\) and \(-7x\)
• Constant terms: \(-12\) and \(+12\)

To combine the variable terms, subtract \(7x\) from \(6x\), resulting in \(-x\). For the constants, adding \(-12\) and \(+12\) cancels them out, leaving us with \(-x = 14\). By combining these like terms, we simplify the equation, making it easier to solve.

Variables and Constants

Understanding variables and constants is crucial when solving equations. A variable is a symbol, commonly \(x\), that represents an unknown value we want to find. A constant is a value that does not change and is not multiplied by a variable. In our exercise, \(x\) is the variable, while numbers like \(6, -2, 7\) in the expression are constants. It’s important to differentiate between them because solving equations usually involves isolating the variable by eliminating the constants through algebraic operations. Recognizing which parts of the equation are variable terms and which are constant terms helps in methods like combining like terms and solving for \(x\).

Solving for x

Solving for \(x\) means finding the value of \(x\) that makes the equation true. After simplifying the equation to \(-x = 14\) by combining like terms, we need to isolate \(x\) to solve for it. In this case, \(-x\) implies \(-1\times x\), so the next step is to clear the negative sign by multiplying both sides by \(-1\):

• The product of \(-1\) and \(-x\) is \(x\)
• On the right side, \(-1\times14 = -14\)

This gives us \(x = -14\), the solution to the equation. Always make sure to perform the same operation on both sides of the equation to maintain its balance. By carefully following these steps, you can solve linear equations securely and steadily every time!