Problem 42 - Algebra 1: Concepts and Skills

Question

Solve \(4.5-7.2 x=3.4 x-49.5 .\) Round to the nearest tenth. You can multiply an equation with decimal coefficients by a power of ten to get an equivalent equation with integer coefficients. Multiply each side of this equation by 10 to rewrite the equation without decimals. $$ \begin{aligned} 4.5-7.2 x &=3.4 x-49.5 \\ 10(4.5-7.2 x) &=10(3.4 x-49.5) \\ 45-72 x &=34 x-495 \\ 45 &=106 x-495 \\ 540 &=106 x \\ \frac{540}{106} &=x \\ 5.094339623 &=x \\ 5.1 & \approx x \end{aligned} $$ The solution is approximately 5.1. Check this in the original equation. Solve the equation. Round to the nearest tenth. $$ 0.625 y-0.184=2.506 y $$

Step-by-Step Solution

Answer

The solutions to the equations are \(x = 5.1\) and \(y = 0.1\).

Step 1: Convert Decimals to Integers

In the first equation, \(4.5 - 7.2x = 3.4x - 49.5\), multiply every term by 10 to get rid of the decimals. This yields the equation \(45 - 72x = 34x - 495\).

Step 2: Rearrange and Solve the Equation

Now bring the terms involving \(x\) on one side and constant terms on the other side of the equation to get \(45+495 = 106x\). This simplifies to \(540 = 106x\). Solving for \(x\), we get \(x = 540/106 = 5.094339623\). But we need to round this to the nearest tenth, so the answer is \(x = 5.1\).

Step 3: Convert Decimals to Integers

For the second equation, \(0.625y - 0.184 = 2.506y\), we multiply every term by 1000 to get rid of the decimals. This yields the equivalent equation \(625y - 184 = 2506y\).

Step 4: Rearrange and Solve the Equation

Rearrange the terms involving \(y\) on one side and constant terms on the other to get \( - 625y + 2506y = 184\). Simplifying, we get \(1881y = 184\), solving for \(y\) gives us \(y = 184/1881 = 0.097810946\). Rounding to the nearest tenth, we obtain \(y = 0.1\).

Key Concepts

Decimal Coefficients

When working with linear equations, you might often come across equations with decimal coefficients, like \(4.5 - 7.2x = 3.4x - 49.5\). These coefficients can make solving equations a bit more challenging, because decimals are harder to work with than whole numbers or integers.
To simplify the problem, it can help to imagine the decimal coefficients as being multiplied by a power of ten. This can convert them into integers. For example, multiplying both sides of the equation by 10 turns \(4.5 - 7.2x = 3.4x - 49.5\) into \(45 - 72x = 34x - 495\). This step does not change the solutions of the equation, as long as every term is multiplied by the same power of ten. Now we have an equivalent equation that is much easier to handle without decimals.

Equivalent Equations

Equivalent equations are equations that have the same solutions, even though they may look different. This concept comes into play when you transform an equation by adding, subtracting, multiplying, or dividing all terms by a non-zero number. These operations do not affect the solution set of the equation.
In our problem, multiplying an equation by 10 changed all decimal coefficients into integer coefficients. The original and transformed equations are equivalent. This means the solution to the integer-focused equation \(45 - 72x = 34x - 495\) will also work for the original equation. This is because the multiplication operation keeps the balance of the equation intact, maintaining the same relationships between the variables and their coefficients.

Rounding Numbers

Once you've solved an equation, sometimes you'll need to round your answer to a specified place value. Rounding numbers involves adjusting a number to make it shorter or simpler, yet still keeping it close in value to the original number.
In our exercise, after solving for \(x\) and finding \(x = 5.094339623\), we needed to round this number to the nearest tenth. To do this, you look at the number in the hundredths place (which is 9 in this case), and if it's 5 or greater, you round up. Therefore, \(x = 5.1\). Rounding is crucial in real-world applications where an exact number may not be necessary or possible, making it easier to communicate and understand.

Integer Coefficients

Integer coefficients in an equation make it easier and simpler to perform algebraic operations compared to decimal coefficients. Working with whole numbers can streamline the solving process because they are straightforward to multiply, divide, add, and subtract.
For instance, in the problem above, after converting decimal coefficients to integer coefficients by multiplying the entire equation by 10, the resulting equation became \(45 - 72x = 34x - 495\). You can clearly see how integer coefficients make the calculation more manageable. Transforming decimal coefficients into integer ones not only helps in algebra but is also widely used in computer science and engineering, where precision is critical and calculations need to be efficient.