Question

What is the solution to the equation? $$ 5(x-2)+x=2(3-x) $$ (A) 2 (B) \(\frac{8}{7}\) (C) \({ }_4\) (D) \(\frac{7}{16}\)

Short answer

The correct solution for the given equation is \(\boxed{\text{(A)}\ 2}\).

Step-by-step solution

Apply the distributive property

First, we need to distribute 5 in the left-hand side of the equation and 2 in the right-hand side of the equation: \(5 \cdot (x - 2) = 5x - 10\) \(2 \cdot (3 - x) = 6 - 2x\) Now the equation becomes: \(5x - 10 + x = 6 - 2x\)

Combine like terms

We will combine the like terms on both sides of the equation: \(5x + x = 6x\) \(-10 = 6 - 2x\) The equation now becomes: \(6x = 6 - 2x + 10\)

Solve for x

Now we will isolate x by adding 2x to both sides of the equation: \(6x + 2x = 8x\) \(8x = 16\) Now, we will divide both sides by 8: \(x = \frac{16}{8}\) \(x = 2\)

Compare with the given options

The obtained solution for x is 2, which matches option (A). Therefore, the correct solution is: \(\boxed{\text{(A)}\ 2}\)

Key concepts

Algebraic Equations

An algebraic equation is a mathematical statement that asserts the equality of two expressions. It is composed of variables and constants, and it typically involves arithmetic operations such as addition, subtraction, multiplication, and division. The primary goal when working with algebraic equations is to find the value of the variable that makes the equation true.

For example, in the equation provided, it is essential to determine the value of 'x' that equates both sides of the equation. Once the variable is isolated and its value determined, the solution can be checked by substituting it back into the original equation to verify if both sides equal.

Distributive Property

The distributive property is a critical algebraic property which allows us to multiply a single term by each term inside a set of parentheses. This is essential for simplifying algebraic expressions and is used when handling algebraic equations that contain brackets.

For instance, in our equation, the distributive property was used to expand the expression on both sides, turning \(5(x-2)\) into \(5x - 10\) and \(2(3-x)\) into \(6 - 2x\). It's a foundational step in solving equations as it simplifies the expressions, paving the way for further manipulation like combining like terms and isolating the variable.

Combining Like Terms

Combining like terms is the process of consolidating terms within an algebraic expression that contain the same variable raised to the same power. This step is crucial in simplifying the equation to ultimately isolate the variable.

In this scenario, 'like terms' refer to terms that have the same variable 'x'. Upon expanding the equation using the distributive property, we add \(5x\) and \(x\) on the left side to get \(6x\), and similarly, we consider the constants separately. The equation simplifies to express all like terms together, which is a strategic move towards finding the value of 'x'.

Isolating Variables

Isolating variables is the process of manipulating the equation so that the variable we need to solve for is by itself on one side of the equation. This involves undoing any addition, subtraction, multiplication, or division performed on the variable.

In the example given, after applying the distributive property and combining like terms, the next step is to isolate 'x'. By adding \(2x\) to both sides and then dividing by 8, the variable 'x' stands alone on one side of the equation. This reveals the solution for 'x', which is then compared with the given options to verify the correct answer. By mastering the technique to isolate variables, one can solve a wide range of algebraic equations efficiently.