Question

Which of the following is equivalent to \(10+2(x-7)\) ? A) \(-14 x+10\) B) \(2 x+24\) C) \(2 x+3\) D) \(2 x-4\) $$ \begin{aligned} & 3 x-\frac{y}{3}=21 \\ & x=y+7 \end{aligned} $$

Short answer

The short answer is: D) \(2x - 4\)

Step-by-step solution

Apply the distributive property

Apply the distributive property by multiplying 2 with both terms inside the parenthesis: \(2(x-7) = 2\cdot x - 2\cdot 7\)

Simplify the expression

Now, we substitute the result from step 1 back into the original expression and perform the multiplication: \(10 + 2x - 14 = 2x -4\)

Compare the result with the given options

By comparing the answer from Step 2 with the given options, we see that the simplified expression matches with option D: \(2x-4\).

Key concepts

Distributive Property

The distributive property is a cornerstone of algebra that allows us to multiply a single term by each term inside a parenthesis. When faced with an expression like \(10 + 2(x - 7)\), the distributive property comes into play to simplify this expression.

What we do here is ‘distribute’ the 2 across the terms within the parenthesis: \(2 \times x\) and \(2 \times -7\). The operation results in \(2x - 14\). Understanding this concept is crucial because it not only makes the equation simpler but also sets the stage for further simplification which is essential in problem-solving.

Improving on the exercise, remember that the distributive property also applies in reverse; this means you can factor out common factors when you encounter expressions like \(2x - 14\). This skill is invaluable in recognizing patterns and simplifying complex expressions.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form while maintaining their value. It means combining like terms, performing arithmetic operations, and applying algebraic rules. Take the expression \(10 + 2x - 14\) for example; after using the distributive property, we simplify by combining the constant terms (10 and -14) to end up with \(2x - 4\).

This step is crucial in problem-solving since it helps make equations more manageable and clearer. In terms of the SAT Math Problem, this step is the bridge that leads to identifying the correct answer among the choices.

When striving for improvement, always ensure that you simplify expressions methodically: combine like terms, perform operations with constants and coefficients, and make sure to double-check for any common factors that can be factored out. Simplification is not just about making an expression shorter; it’s about making it more understandable and easier to work with.

Equivalent Expressions

Equivalent expressions are different expressions that have the same value for all values of the variables involved. In the context of the SAT example, we are trying to determine which option is equivalent to the original expression \(10+2(x-7)\). After applying the distributive property and simplifying, we found the expression to be \(2x-4\).

We compare this simplified expression to the answer choices and recognize that option D, \(2x-4\), is indeed equivalent to the original expression. This illustrates the concept well—as different as two expressions might look at first glance, they can represent the same relationship or value.

In exercises, remember that checking for equivalency often involves simplification first. Only when expressions are in their simplest form can we accurately determine whether they are equivalent or not. This concept is not just important for standardized tests like the SAT, but also for understanding more complex relationships in algebra and beyond.