Problem 27 - Algebra 1: Concepts and Skills

Question

Solve the equation. Round the result to the nearest hundredth. Check the rounded solution. $$ -35 m+75=48 $$

Step-by-Step Solution

Answer

After solving the equation and rounding off to the nearest hundredth, \( m = 0.77 \)

Step 1: Isolate the term with m on one side of the equation

To isolate \( m \), first get rid of the constant on the left side of equation. To do this, subtract 75 from both sides of the equation. This will result in a new equation: \( -35m = 48 - 75\)

Step 2: Simplify the equation

Simplify the right side of the equation by performing the subtraction. \( -35m = -27 \)

Step 3: Solve for m

To solve for \( m \), you divide both sides of the equation by -35. This will isolate \( m \) on one side of the equation. Hence, \( m = \frac{-27}{-35} \).

Step 4: Round the answer

Calculate the division and round the result to the nearest hundredth. This gives \( m = 0.77 \).

Step 5: Check the solution

Insert \( m = 0.77 \) in the original equation and verify if the equation holds true. If so, then m=0.77 is indeed the correct answer.

Key Concepts

Isolation of Variables

When solving linear equations, one of the primary goals is to isolate the variable on one side of the equation. This means getting the variable, in this case, \( m \), all by itself. In the original problem, the equation is \(-35m + 75 = 48\). The term containing the variable \( -35m \) needs to be isolated.To do this, we first need to remove any constants from the side of the equation that has the variable. A constant, like \( 75 \), does not contain a variable. To remove it, we perform the inverse operation—in this instance, subtraction. Thus, by subtracting 75 from both sides, we begin isolating \( m \). This results in a simpler equation: \(-35m = 48 - 75\).Here are some steps to remember when isolating a variable:

• Identify the term with the variable.
• Perform the inverse operation to remove constants.
• Ensure operations are done on both sides to maintain equality.

Isolation prepares us for the next steps of solving the equation.

Equation Simplification

Simplification of an equation makes it easier to solve and understand. Once we have isolated the variable term, the next step is to simplify the expressions on each side of the equation. In practice, this often involves basic arithmetic like addition, subtraction, multiplication, or division.For the equation \(-35m = 48 - 75\), we simplify the right side by performing the subtraction of these numbers: \(48 - 75 = -27\). This simplifies our equation to \(-35m = -27\).Simplification usually involves:

• Performing operations like addition or subtraction with constants.
• Simplifying fractions, if necessary.
• Reducing terms to their simplest forms.

Once simplified, we can more clearly see how to solve for the variable.

Rounding Numbers

Rounding is a mathematical practice aimed at simplifying numbers to make them easier to work with or communicate. In solving equations, rounding often comes into play after calculations have provided a raw, potentially complex result.In our problem, after isolating and solving for \( m \), we calculated that \( m = \frac{-27}{-35} \). This division gives \( m \approx 0.77142857\ldots \), a repeating decimal. To simplify this, we round to the nearest hundredth, resulting in \( m = 0.77 \).Important points for rounding:

• Determine the place value to round to, such as hundredths.
• If the next digit (to the right) is 5 or greater, round up.
• If it is less than 5, round down by leaving the place value as is.

Once rounded, it is useful to check the solution with the original equation to verify accuracy.