Problem 35 - Algebra A Combined Function

Question

Solve each equation. Don't forget to first simplify each side of the equation, if possible. Check each solution. See Examples 5 through 7 . $$ -2(x+6)+3(2 x-5)=3(x-4)+10 $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 25\).

Step 1: Distribute

Apply the distributive property to open up the brackets. This means multiplying each term inside a pair of parentheses by the factor outside.For the left side:\[-2(x+6) = -2x - 12\] and \[3(2x - 5) = 6x - 15\]For the right side:\[3(x - 4) = 3x - 12\]This transforms the equation to:\[-2x - 12 + 6x - 15 = 3x - 12 + 10\]

Step 2: Combine Like Terms

Simplify both sides by combining like terms.On the left side:\[-2x + 6x = 4x\] and \[-12 - 15 = -27\]On the right side:\[-12 + 10 = -2\]This gives the simplified equation:\[4x - 27 = 3x - 2\]

Step 3: Isolate the Variable

Get all terms with the variable \(x\) on one side and constant terms on the other.Subtract \(3x\) from both sides:\[4x - 3x - 27 = 3x - 3x - 2\]This simplifies to:\[x - 27 = -2\]Add 27 to both sides to isolate \(x\):\[x - 27 + 27 = -2 + 27\]This simplifies further to:\[x = 25\]

Step 4: Check the Solution

Substitute \(x = 25\) back into the original equation to verify.Original equation:\[-2(x+6) + 3(2x-5) = 3(x-4) + 10\]Substituting \(x = 25\):\[-2(25+6) + 3(2(25)-5) = 3(25-4) + 10\]Calculate:\[-2(31) + 3(50-5) = 3(21) + 10\]\[-62 + 3 imes45 = 63 + 10\]\[-62 + 135 = 73\]\[73 = 73\]Since both sides equal, the solution \(x = 25\) is verified.

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra. It allows us to simplify expressions where a number multiplies a sum or difference inside parentheses. This property states that for any numbers, say, a, b, and c, the following equation holds true: \[ a(b + c) = ab + ac \]When you see this kind of expression in an algebraic equation, you'll want to "distribute" the outside number by multiplying it with each term inside the parentheses. It is like spreading the number over each term.In our example, we have \[-2(x+6) + 3(2x-5)\] on one side.

• First, distribute \(-2\) over \((x + 6)\) resulting in \(-2x - 12\).
• Then multiply \(3\) by \(2x - 5\) to get \(6x - 15\).

This same approach is applied to any expression with parentheses in the equation. Remembering this property will help you tackle complex problems with ease.

Combining Like Terms

Once you have applied the distributive property, the next step in simplifying an algebraic equation is to combine like terms. Like terms are terms in the equation that have the same variable raised to the same power. For instance, terms like \(2x\) and \(3x\) are like terms because they both contain the variable \(x\).In simplifying expressions, grouping these terms together helps make the equation more manageable and clearer.

• On the left side of our equation, combine \(-2x\) with \(6x\), which simplifies to \(4x\).
• Also combine the numerical terms, \(-12\) and \(-15\), simplifying to \(-27\).

This process results in a neater and often simpler expression to solve.The aim of combining like terms is to reduce the equation to its simplest form, enabling easier manipulation and eventual solution.

Solving Equations

Solving equations is the ultimate goal in algebra. The aim is to find the value of the variable that makes the equation true. Once the equation is simplified using the distributive property and combining like terms, the next step is solving the equation typically by isolating the variable.Let's take our simplified equation \(4x - 27 = 3x - 2\) and solve it:

• First, bring all terms including the variable \(x\) to one side. Subtract \(3x\) from both sides, resulting in \(x - 27 = -2\).
• Next, move the constant term on the side with \(x\) by adding \(27\) to both sides, to end with \(x = 25\).

It's always crucial to check your solution by substituting the value back into the original equation, ensuring both sides equal. This process confirms the correctness of your solution. Solving equations may seem challenging, but by following the steps methodically, you can efficiently find the right solution.