Problem 50 - Algebra 1

Question

Evaluate the expression for the given value of x. $$9+x+(-8)+(-3) ; x=-12$$

Step-by-Step Solution

Answer

The evaluated expression for x=-12 is -14.

Step 1: Substitute x into the expression

Replace the variable x in the expression \(9+x+(-8)+(-3)\) with the given value \(x=-12\). This gives us \(9 + (-12) + (-8) + (-3)\).

Step 2: Combine Like Terms

Now, combine all the constants in the equation. We add 9, -12, -8, and -3. The sum of these values is \(-14\).

Step 3: Conclusion

So, the result of evaluating the expression \(9+x+(-8)+(-3)\) for the value \(x=-12\) is \(-14\).

Key Concepts

Substituting Values

Substituting values into algebraic expressions is akin to solving a mystery, where each variable holds a clue. To reveal the unknown, we replace each variable with a given number. This process is the cornerstone of finding the value of an expression. In the context of our exercise where we have the expression \(9+x+(-8)+(-3)\), and the variable \(x\) is given the value \(-12\), we essentially swap every \(x\) with \(-12\). It's like exchanging puzzle pieces until the full picture is visible: \(9 + (-12) + (-8) + (-3)\).

For students tackling this for the first time, think of it as knowing the price of an item and the amount you have. If the item is 'x' dollars and you have a $12 coupon, you 'substitute' the value of the coupon when calculating how much you'll spend.

Combining Like Terms

Picture combining like terms as if you're organizing a toolbox. In a disorganized toolbox, you might have screws, nails, and bolts scattered all over. When you combine like terms, you're sorting out these items and grouping them together. This simplifies counting them, just as we simplify expressions in algebra. In our exercise, the constants (numbers without variables) are like different types of hardware. We have 9, -12, -8, and -3, all of which can be summed because they're like terms.

The calculation here is straightforward: \(9 + (-12) + (-8) + (-3) = -14\). Through this process of simplification, we tidy up the equation and make it easier to understand. It's a critical step that often makes the difference between a correct or incorrect answer.

Algebraic Expressions

Algebraic expressions are the phrases of the mathematical world, consisting of numbers, variables, and operations that tell a numerical story. An expression is like a recipe, with variables as ingredients and mathematical operations as cooking instructions. The expression \(9+x+(-8)+(-3)\) invites us to mix these ingredients following the mathematical recipe. Expressions can be short and simple or long and complex, but they don't include an equals sign like equations do.

It's essential to understand that the value of an expression changes depending on the variable's value, which is why substituting values is a significant first step in evaluating them.

Simplifying Expressions

Simplifying an expression is to algebra what decluttering is to your bedroom; it means to streamline the expression into its simplest form. The goal is to make it as neat and as easy to read as possible, with as few terms as necessary. Simplification may involve combining like terms, using the distributive property, and reducing fractions.

In our problem, after substituting \(x\) with \(-12\), we simplify by adding the constants: \(9 + (-12) + (-8) + (-3) = -14\). Think of simplification as cleaning up after cooking; once you’ve followed the recipe, you clean up to leave a neat and tidy kitchen—or in this case, a neat and tidy mathematical answer.