Problem 21 - Algebra A Combined Function

Question

Solve each equation. See Examples 3 through \(5 .\) $$ 0.50 x+0.15(70)=35.5 $$

Step-by-Step Solution

Answer

The solution is \(x = 50\).

Step 1: Simplify the constant term

Calculate the value of the constant term in the equation. In this case, find \(0.15 \times 70\) which simplifies to \(10.5\). Now, rewrite the equation as \[ 0.50x + 10.5 = 35.5 \]

Step 2: Isolate the variable term

Subtract \(10.5\) from both sides of the equation to isolate the term containing \(x\). This gives\[ 0.50x = 35.5 - 10.5 \] which further simplifies to \[ 0.50x = 25 \].

Step 3: Solve for the variable

Divide both sides of the equation by \(0.50\) to solve for \(x\): \[ x = \frac{25}{0.50} \] This gives \[ x = 50 \].

Key Concepts

Constant Term

In any algebraic equation, a constant term is a number that stands alone without any variable attached to it. When dealing with linear equations, identifying and operating on constant terms is crucial.

• In our example, after calculating, the constant term present in the equation is 10.5.
• The equation thus reads \[ 0.50x + 10.5 = 35.5 \]When you identify a constant term, make a note of the mathematical operations that you might need to perform on it; this most often includes addition or subtraction, multiplications as seen here.

Breaking down calculations involving constant terms step by step can help prevent errors. Always double-check your multiplications. In this exercise, we calculated that \[ 0.15 \times 70 = 10.5 \].Understanding and calculating constant terms simplifies the process of solving equations.

Isolate Variable

Isolating a variable means rearranging an equation so that the variable you're solving for is on one side, and everything else is on the other. This concept is key for solving equations.

• For the example here, the variable to isolate is \(x\), present in the term \(0.50x\).
• To start, subtract the constant term \(10.5\) from both sides to keep the equation balanced. The result will be\[ 0.50x = 35.5 - 10.5 \]
• Simplify the subtraction to obtain \[ 0.50x = 25 \]

The process of isolating variables often requires careful arithmetic operations like addition, subtraction, or division. Techniques might vary, but the ultimate goal remains the same: make \(x\) easily identifiable on one side of the equation.

Solve Equation

Solving an equation means finding the numerical value of the variable that makes the equation true. This is the core aspect of working with linear equations.

• In our example, we simplify from \( 0.50x = 25 \) to solve for \(x\).
• Divide both sides by \(0.50\) to isolate \(x\):\[ x = \frac{25}{0.50} \]
• This calculation gives us \[ x = 50 \]

If done properly, solving the equation leads you to the value of the variable, achieving the solution. A correct approach to division, especially involving decimals, is essential to acquire the correct answer, like we did here.

Simplify Expression

Simplifying an expression involves performing operations to make the equation easier to solve while maintaining equality.

• Initially, simplifying means evaluating the multiplication \[ 0.15 \times 70 = 10.5 \]
• The equation then is simplified into \[ 0.50x + 10.5 = 35.5 \]Following subtraction of constants, further simplifies to \[ 0.50x = 25 \]

By simplifying, we reduce complexity, allowing us to apply straightforward operations like division to find the solution more efficiently. Review all simplification steps to ensure accuracy and clarity. In this way, you maintain the balance of the equation while making it easier to manage.