Problem 63 - Algebra 1: Concepts and Skills

Question

Solve the equation. Round the result to the nearest hundredth. $$10.52 x+1.15=-1.12 x-6.35$$

Step-by-Step Solution

Answer

The solution to the equation is \(x=-0.64\).

Step 1: Simplify the Equation

Start by moving all terms with \(x\) on one side and the constants on the other side. By adding \(1.12x\) to both sides and subtracting \(1.15\) from both sides, the equation simplifies to:\[10.52x+1.12x = -6.35 - 1.15\]

Step 2: Combine Like Terms

Then combine the like terms which gives:\[11.64x = -7.5\]

Step 3: Solve for \(x\)

Divide both sides of the equation by the coefficient of \(x\), which is \(11.64\), we have:\[x = \frac{-7.5}{11.64}\]

Step 4: Round to the Nearest Hundredth

After calculating the above value, round the result to two decimal places.

Key Concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like x or y), and operators (such as plus, minus, multiply, and divide). For instance, in the given exercise \(10.52x + 1.15=-1.12x - 6.35\), the equation is an algebraic expression featuring variables (x), constants (1.15 and -6.35), coefficients (10.52 and -1.12 which are the numbers before the x), and the equality sign.

Understanding the structure of algebraic expressions is vital because it helps in simplifying and solving equations. For example, recognizing which parts of the expression are like terms or how to move terms from one side of the equality to the other is the foundation of algebraic manipulation. Through practice and application of these concepts, one can become more proficient in solving complex algebraic equations.

Combining Like Terms

Combining like terms is a crucial step in solving algebraic equations. Like terms are terms that contain the same variable or variables raised to the same power. In our exercise, once we have moved all terms containing x to one side, we encounter like terms: \(10.52x\) and \(1.12x\).

Here's a simple guideline on how to combine like terms:

• Identify terms that have the exact variable component.
• Add or subtract the coefficients of these terms as indicated in the equation.
• Maintain the variable part unchanged.

In this case, when we add the coefficients (10.52 and 1.12), we get a new coefficient of 11.64 for the like terms, resulting in the simplified term \(11.64x\). This simplified form makes it much easier to solve for the variable.

Rounding Decimals

Rounding decimals is a numerical process used to reduce the number of digits right of the decimal while keeping the value approximately the same. It's particularly useful in providing a more manageable figure when exactness is unnecessary, or when we need to comply with specified precision requirements. In our exercise, we are asked to round to the nearest hundredth, which means keeping two decimal places.

Steps for Rounding to the Nearest Hundredth:

• Identify the digit at the hundredth place (two places to the right of the decimal point).
• Look at the next digit (thousandth place); if it's 5 or more, increase the hundredth place by one.
• If the next digit is less than 5, leave the hundredth place as it is.
• Drop all digits after the hundredth place.

By applying these steps, we ensure numerical values are practical and suited for interpretation or further calculations.