Question

Solve the equation. Round the result to two decimal places. $$15.67 x+23.61=1.56+45.8 x$$

Short answer

The solution to the equation is approximately \(x = -0.73\).

Step-by-step solution

Rearrange terms

Gather all terms with \(x\) on one side and the constants on the other. We are left with \( 30.13x = -22.05 \)

Isolate x

To isolate \(x\), we can divide both sides of the equation by 30.13: \( x = \frac{-22.05}{30.13} \)

Calculate and round off

Now calculate the right side and round the result to two decimal places: \(x \approx -0.73\)

Key concepts

Understanding Linear Equations

Linear equations form the foundation of algebra and are used to describe relationships between variables with a constant rate of change. These equations are easily identified by their characteristic format, resembling the standard formula: \(ax + b = c\), where \(a\), \(b\), and \(c\) represent constants, and \(x\) is the variable we want to solve for. The ability to manipulate linear equations is crucial for solving various real-world problems, from calculating distances to predicting profits.

In the given exercise, the equation presented is a textbook example of a linear equation where you have two terms with the variable \(x\) and constants on both sides of the equation. To solve this, the first step is simplifying and consolidating all variable terms on one side and constant terms on the opposite side. Here you should always watch out for the signs that precede the numbers, as they are critical in determining the correct equation to ultimately solve for \(x\).

Isolating the Variable

Once the linear equation is simplified, the next vital concept is to isolate the variable. 'Isolating the variable' means rewriting the equation so that the variable stands alone on one side of the equality, with all other terms on the opposite side. This is typically achieved through the inverse operations of addition, subtraction, multiplication, or division.

In the context of our exercise, after consolidating like terms, we divide both sides by the coefficient of \(x\), which is \(30.13\), to get the variable by itself. It's essential to perform this operation on both sides to maintain the balance of the equation—what's done to one side must be done to the other. The division will nullify the \(x\)'s coefficient on the left and leave us with the isolated variable \(x\) on the left side, with its solution on the right side.

Rounding Decimals

Rounding decimals is a significant concept, especially when dealing with real-world quantities where exact numbers are not always necessary, or when an approximate value is preferable for simplicity. To round to a specific decimal place, you look at the number in the next decimal place to the right; if this number is five or higher, you increase the previous number by one. If it's less than five, the previous number stays the same.

Once the value of \(x\) is computed, we adhere to the exercise's instruction to round the result to two decimal places. For instance, if we calculated \(x\) and got an approximate value of -0.731, since the third decimal place (1) is less than five, we would round down, and the two-decimal approximation becomes -0.73. It's a helpful technique to simplify answers, making them easier to understand and utilize in further calculations or practical applications.