Question

Solve the equation. $$3 x-7+x=5$$

Short answer

The solution of the equation is \(x = 3\).

Step-by-step solution

Simplify Like Terms

Combine the \(x\) terms on the left side of the equation. This results in the equation \(4x - 7 = 5\)

Isolate the x term

Move the -7 from the left side to the right side of the equation by adding 7 to both sides. Resulting in the equation \(4x = 12\)

Solve for x

Now, to isolate \(x\) we divide both sides of the equation by 4. This results in the solution \(x = 3\)

Key concepts

Combining Like Terms

When you are presented with an algebraic equation, such as \(3x - 7 + x = 5\), the first step is typically to combine like terms. Like terms are terms that have the same variable raised to the same power. In this case, both \(3x\) and \(x\) are like terms because they both represent some multiple of \(x\). To combine them, you simply add or subtract the coefficients (the numbers in front of the variables). So \(3x + x\) becomes \(4x\), which simplifies the original equation to \(4x - 7 = 5\).

Understanding how to combine like terms is crucial for simplifying equations and making them easier to solve. Remember to look for terms that have the same variable and combine them by performing basic arithmetic with their coefficients.

Isolation of Variables

The essence of solving an equation is to isolate the variable; in other words, you want to get the variable by itself on one side of the equation. This technique involves inverse operations. For the equation \(4x - 7 = 5\), we need to move the \(7\) to the other side of the equation by adding \(7\) to both sides, according to the principle 'what you do to one side of the equation, you must do to the other.' This gives us \(4x = 12\). The purpose of such manipulation is to have the variable 'x' on one side and a numerical value on the other, leading to a clearer path towards the solution. Isolating the variable is a fundamental step in the process of solving linear equations and is critical for advancing in algebra.

Solving Algebraic Expressions

Once we've isolated the variable to one side of the equation, as in \(4x = 12\), the next step is to solve for the variable. This is where your algebraic manipulation skills come into play. To solve for \(x\), we need to perform operations that will leave \(x\) by itself. In this example, since \(x\) is multiplied by \(4\), the inverse operation is division. Dividing both sides of the equation by \(4\) gives us \(x = 3\), which is the solution to the original problem.

Solving algebraic expressions generally involves a series of these inverse operations until the variable is by itself. Being fluent in identifying and applying the correct inverse operations (addition vs. subtraction, multiplication vs. division) is key to finding the solution to algebraic equations efficiently.