Problem 14 - Algebra and Trigonometry

Question

Solve and check each linear equation. $$5 x-(2 x+2)=x+(3 x-5)$$

Step-by-Step Solution

Answer

The solution for the equation \(5 x-(2 x+2)=x+(3 x-5)\) is \(x = 3\)

Step 1: Collect like terms

On the left-hand side there is \(5x-(2x+2)\) and on the right-hand side \(x+(3x-5)\). After distributing the negative sign on the left-hand side, you get \(5x - 2x - 2\) and after combining like terms on each side, this simplifies to \(3x - 2\). Similarly, on the right-side, combining like terms \(x + 3x\) gives \(4x - 5\). So, the equation becomes \(3x - 2 = 4x - 5\).

Step 2: Move terms involving \(x\) to one side and constants to the other

To help isolate the \(x\) value, it's useful to start by moving all \(x\) terms to one side and the constants to the other. By subtracting \(3x\) from both sides, you get \(-2 = x - 5\). And then adding \(5\) to both sides gives \(x = 3.\)

Step 3: Verify the solution

Place the obtained value of \(x\) back to the original equation. Therefore, substituting \(x = 3\) back into the equation, you get \(5(3) - (2(3) + 2) = 3 +(3(3) - 5)\) which simplifies to \(15 - 8 = 3 + 4.\) Both sides simplify to \(7\), hence verifying that \(x = 3\) is the solution.

Key Concepts

Collecting Like Terms

When solving linear equations, one of the fundamental steps is collecting like terms. Like terms are terms that have the same variable raised to the same power. The coefficients of these terms can be different. For instance, in the given equation, terms containing the variable 'x' on each side are like terms:

To effectively combine them, we must simplify the equation by adding or subtracting these terms. This process makes the equation easier to solve because it reduces the number of terms we're working with. For example, in the equation given, merging like terms on the left-hand side gives us \(5x-(2x+2)\) which simplifies to \(3x - 2\), and on the right, \(x+(3x-5)\) simplifies to \(4x - 5\). It's like tidying up a messy room by grouping similar items together—you get a clearer picture of what you're dealing with.

Distributive Property

The distributive property is a powerful tool in algebra that allows us to multiply a single term by each term inside a set of parentheses. This is crucial when you encounter expressions like \(5x-(2x+2)\). To clear the parentheses, we distribute the negative sign across \(2x+2\), effectively changing the signs of each term within the parentheses. This process gives us \(5x - 2x - 2\).

It's important to remember that every term inside the parentheses gets multiplied by the factor outside. Don't forget the sign—whether positive or negative—as it significantly affects your results. By applying this property, we're setting the stage for collecting like terms, which further simplifies the expression.

Isolating Variables

The key to solving a linear equation is isolating the variable so that it stands alone on one side of the equation. This process often involves moving terms that contain the variable to one side and constants to the other. For example, in our problem, we aim to isolate 'x'.

To achieve this, we subtract \(3x\) from both sides to remove the 'x' from the right side of the equation, resulting in \(x = 3\). Remember, whatever operation you do to one side, you must also do to the other to keep the equation balanced. Think of it like a seesaw: to keep it level, the weight (or in our case, the numerical value) needs to be distributed evenly.

Verifying Solutions

After arriving at a potential solution for our variable, it's not the end. We need to ensure that our solution is correct by verifying the solution. We do this by substituting the value back into the original equation to see if it holds true. In essence, if we've solved the equation accurately, plugging in our 'x' value will yield equal values on both sides of the equation.

In our problem, substituting \(x = 3\) back into the original equation should result in identical expressions on both sides, confirming that we've found the correct solution. It's akin to trying a key in a lock—if it turns smoothly and the door opens, we know we've got the right key. Similarly, the equation 'opening up' to reveal the same numbers on both sides confirms a successful solution.