Problem 12 - Algebra and Trigonometry

Question

Solve and check each linear equation. $$2-(7 x+5)=13-3 x$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 1.6\).

Step 1: Simplify Both Sides

First, simplify the expressions on both sides of the equation if possible. In this case, the equation already simplified is given, so this step doesn't add anything to our solution.

Step 2: Collect Like Terms

Move all terms involving the variable \( x \) to one side of the equation, and all the constants to the other. Doing this we get: \(7 x + 3 x = 13 - 2 + 5\).

Step 3: Simplify

Simplify the equation by performing actual addition or subtraction:\(10x = 16\).

Step 4: Solve for \(x\)

Lastly, divide both sides by the coefficient of \( x\) (which is 10 in this case) to isolate and solve for \( x \). Doing this we get:\(x = \frac{16}{10}\).Simplify the fraction and we get \(x = 1.6\).

Key Concepts

Linear Equation

A linear equation is a type of equation that makes a straight line when graphed. These equations are usually written in the form of ax + b = c, where a, b, and c are constants, and x is the variable we want to solve for. They're called 'linear' because they represent relationships where the variables change at a constant rate. When solving these equations, our goal is to find the value of the variable that makes the equation true.

In our exercise, we have the linear equation 2 - (7x + 5) = 13 - 3x, which after simplifying, leads to finding the value of x that satisfies this condition. Linear equations are foundational in algebra and are used to model various real-world situations.

Isolate Variable

To isolate the variable means to get the variable by itself on one side of the equation. Doing this allows you to solve the equation and find the value of the variable. In most cases, you'll perform a series of operations that are the opposite (inverse operations) of what's currently affecting the variable. For instance, if a variable is multiplied by a number, you'd divide both sides by that number to isolate it.

In our example, by the time we reach step 4, we have 10x = 16. To isolate x, we divide both sides by 10, giving us x = 16/10 or x = 1.6 after simplification. Remember, whatever you do to one side of the equation, you must also do to the other to maintain the balance.

Collect Like Terms

When we collect like terms, we're combining terms that have the exact same variable raised to the same power. For example, 3x and 7x are like terms because they both have the variable x and can be combined to get 10x. Collecting like terms helps to simplify the equation and get us closer to isolating the variable.

In our exercise, we combined the terms 7x and -3x on the left side to get 10x and combined the constants on the right side to simplify our equation. This process is essential for solving linear equations efficiently.

Simplify Expressions

To simplify expressions means to perform all the possible arithmetic operations and to reduce the expression to its simplest form. This can involve expanding brackets, combining like terms, and wherever possible, reducing fractions to their lowest terms. The key is to eliminate any unnecessary complexity that might make the equation harder to solve.

After collecting like terms in the provided exercise, we simplified the expression 10x = 16 to x = 1.6 by dividing both sides by 10, which gives us the simplest form of the solution. Simplifying expressions not only helps in solving equations but also in understanding the structure and potentially revealing more insights into the problem at hand.