Problem 18 - Algebra 1: Concepts and Skills

Question

Solve the equation. Round the result to the nearest hundredth. Check the rounded solution. $$ 2.69-3.64 x=23.78 x $$

Step-by-Step Solution

Answer

The value of \(x\), rounded to the nearest hundredth, is approximately 0.10

Step 1: Simplify the equation

Move all terms with \(x\) on one side by adding \(3.64x\) to both sides of the equation: \(2.69 = 23.78x + 3.64x\)

Step 2: Combine like terms

Add \(23.78x\) and \(3.64x\) together: \(2.69 = 27.42x\)

Step 3: Isolate the variable \(x\)

Divide both sides by \(27.42\) to find the value of \(x\): \(x = 2.69 / 27.42\)

Step 4: Round the result

After performing the division, round the value of \(x\) to the nearest hundredth to get the final result. It's crucial to use the correct rounding method here.

Step 5: Check the solution

Substitute \(x\) back into the original equation to verify that both sides equal. This step will ensure the solution is correct.

Key Concepts

Understanding Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. Variables are symbols, often letters like x or y, that represent unknown values. Operations in algebraic expressions include addition, subtraction, multiplication, and division.

Take for instance, in the given exercise, we have the algebraic expression 2.69 - 3.64x = 23.78x. This includes the numbers 2.69, -3.64, and 23.78; the variable x; and the operations of subtraction and multiplication. To solve for x, you need to perform a series of operations that will manipulate the equation to find the value of this unknown variable.

When simplifying algebraic expressions, it's critical to combine like terms, which are terms that have the same variable raised to the same power. For example, 23.78x and 3.64x are like terms in the exercise. They can be combined by addition or subtraction, depending on their coefficients (the numbers multiplying the variables). The goal is to simplify the expression to make the process of isolating the variable straightforward.

Isolating the Variable

The process of isolating the variable involves manipulating the equation so that the variable you're solving for is by itself on one side of the equal sign. The other side of the equation will contain the numbers or expressions that describe its value.

In our equation, we want to isolate x. This means getting x on one side and everything else on the other side. Here's how to do it:

• Add 3.64x to both sides to move all terms containing x to one side.
• Combine the like terms with x to simplify the expression. This changes the equation to 2.69 = 27.42x.
• Finally, divide both sides by 27.42 to solve for x. This leaves x by itself.

The goal is to use inverse operations to get the variable alone. Addition is undone by subtraction, multiplication is undone by division, and vice versa. In this case, since x was multiplied by 27.42, we undid this by dividing by 27.42.

Rounding Decimals

When dealing with real-world problems, sometimes exact numbers aren't necessary, and rounding decimals to a specific place value can give a sufficient approximation. Rounding can reduce the complexity of the results and make them easier to understand or use.

To round a decimal to the nearest hundredth means to keep two digits after the decimal point. Number lines, estimation skills, and understanding place value are all fundamental when rounding. If the third digit after the decimal is 5 or greater, you increase the second digit by one. If it's less than 5, you leave the second digit as it is.

In our exercise, after dividing 2.69 by 27.42 we get a decimal. This result must then be rounded to the nearest hundredth. Proper rounding is essential not just for obtaining a practical answer but also for ensuring accuracy within the required precision.