Problem 28 - Algebra 1: Concepts and Skills

Question

Solve the equation. $$ 5(-x+2)=-3(7 x+2)+8 x $$

Step-by-Step Solution

Answer

The solution for the given equation is \(x = -2\).

Step 1: Apply distributive property to eliminate parentheses

Apply the distributive property \(a(b+c) = ab + ac\) to get:\[5 \cdot (-x) + 5 \cdot 2 = -3 \cdot (7x) -3 \cdot 2 + 8x\]This simplifies to:\[-5x + 10 = -21x -6 + 8x\]

Step 2: Combine like terms

On the right side, combine like terms \(-21x\) and \(8x\) to simplify the equation:\[-5x + 10 = -13x - 6\]

Step 3: Isolate the variable \(x\)

To isolate the variable \(x\), add \(13x\) to both sides of the equation. Also, subtract \(10\) from both sides to get:\[8x = -16\]

Step 4: Solve for \(x\)

To determine the value of \(x\), divide both sides by \(8\), which results to \(x = -2\).

Key Concepts

Distributive Property

Understanding the distributive property is crucial when solving linear equations, especially when they involve parentheses. In essence, this property allows us to remove parentheses by evenly distributing a factor across the terms inside the parentheses. For instance, the expression \(a(b+c)\) can be rewritten as \(ab + ac\). Let's look at how this applies to our exercise:

Initially, we have the equation \(5(-x+2)\). By applying the distributive property, we multiply \(5\) by both \(–x\) and \(2\), resulting in \(–5x + 10\). On the right side of the equation, we do a similar process to distribute \(–3\) across \(7x + 2\), leading to \(–21x – 6\), and then we have a standalone term, \(8x\), to incorporate next.

Through the distributive property, we transform an equation with parentheses into a simplified version that we can work with, setting the stage for combining like terms.

Combine Like Terms

Once the parentheses have been eliminated, the next step in the solution involves combining like terms. Like terms are terms that contain the same variables raised to the same power. For instance, in our equation \(–5x + 10 = –21x – 6 + 8x\), the terms \(–21x\) and \(8x\) both contain the variable 'x' and can be combined.

To combine them, simply add their coefficients: \(–21 + 8\) gives us \(–13x\). Now, the equation has been simplified to \(–5x + 10 = –13x – 6\). Combining like terms effectively streamlines the equation, reducing clutter, and paving the way for us to isolate the variable we are solving for.

Isolate Variables

Isolating the variable is a key step in solving an algebraic equation. It involves moving all instances of the variable to one side of the equation, and all constants to the other. In our problem, after combining like terms, we must isolate 'x'.

We add \(13x\) to both sides, which cancels out the \(–13x\) on the right side and adds to \(–5x\) on the left, giving us \(8x\). At the same time, we subtract \(10\) from both sides to move the constant to the right side. The result is a simplified equation \(8x = –16\), where 'x' is now isolated and ready to be solved for.

Algebraic Solution Steps

The steps taken to solve an algebraic equation can be thought of as a methodical dance, where each move is deliberate and essential to reaching the solution. Our linear equation has gone through several such steps: applying the distributive property, combining like terms, and isolating the variable.

Once we have isolated 'x', the last step is simple. We have \(8x = –16\); to find the value of 'x', we divide both sides by the number in front of 'x', which is \(8\) in this case. This leaves us with \(x = –16 ÷ 8\), which simplifies to \(x = –2\). The methodical approach of following these algebraic solution steps guarantees that we arrive at the correct answer systematically.