Problem 90 - Algebra A Combined Function

Question

Fill in the blank with a number of your choice so that each equation has the given solution. ______ \(x=10 ;\) solution: \(\frac{1}{2}\)

Step-by-Step Solution

Answer

The blank should be filled with \(\frac{1}{20}\).

Step 1: Understand the problem

We need to fill in the blank with an appropriate coefficient for \(x\) so that when the equation is solved, \(x = 10\) gives the solution \(\frac{1}{2}\). This implies substituting \(x = 10\) into the equation should yield \(y = \frac{1}{2}\).

Step 2: Set up the equation

Let's consider the equation \( ax = \frac{1}{2} \) where \(a\) is the number to fill in. The solution implies finding \(a\) such that \(a \cdot 10 = \frac{1}{2}\).

Step 3: Solve for the missing number

From the equation \( a \cdot 10 = \frac{1}{2} \), we solve for \(a\). By dividing both sides by 10, we find \(a = \frac{1}{20}\).

Step 4: Verify the solution

Substitute \(a = \frac{1}{20}\) back into the equation and check: \( \frac{1}{20} \times 10 \) should equal \(\frac{1}{2}\). Simplifying the left side gives \( \frac{10}{20} = \frac{1}{2} \), confirming our solution is correct.

Key Concepts

Solving Equations

Solving equations is a fundamental skill in algebra that involves finding the value of a variable that makes an equation true. Imagine an equation like a balance scale. When you solve it, you are trying to find the perfect balance by determining the value of the unknown variable.
In our exercise, the equation was set up as \( ax = \frac{1}{2} \), and we were tasked with finding the coefficient \(a\) such that when \(x = 10\), the equation holds true. The process of solving this involves isolating the unknown. Here, we isolated \(a\) by dividing both sides by 10:

• Start with \( a \cdot 10 = \frac{1}{2} \).
• Divide both sides by 10 to solve for \(a\): \(a = \frac{1}{20}\).

Through these steps, the equation is solved, providing clarity on the value that satisfies the given condition. Remember, solving equations is all about keeping the equation balanced and systematically isolating the variable.

Coefficients

In algebra, coefficients are numbers that multiply variables in an equation. They give weight to the variables they accompany and essentially determine how impactful each variable is within the equation. The coefficient acts like a scale factor.
In the original problem, the coefficient \(a\) is what we needed to find to make the equation true. The equation \( ax = \frac{1}{2} \) helped us understand that for the given solution, \(x\) required a specific coefficient. With \(x = 10\), we were searching for the number that, when multiplied by 10, results in \(\frac{1}{2}\):

• Define the equation with the variable \(ax = \frac{1}{2}\).
• Determine the coefficient by arranging terms to solve for \(a\).
• Calculate \( a = \frac{1}{2} \div 10 = \frac{1}{20} \).

The coefficient in an equation allows you to directly manipulate the expression to explore different values and their effects, making them essential in algebraic expressions and problem-solving.

Substitution Method

The substitution method is a crucial tool in algebra for solving equations. By replacing a variable with a specific value, you can verify if it satisfies the equation, simplifying the process of finding solutions or checking their correctness.
In this exercise, we used the substitution method to confirm our found coefficient. We substituted \(x = 10\) and our calculated coefficient \(a = \frac{1}{20}\) back into the equation to see if it holds true:

• Set the original equation \( ax = \frac{1}{2}\).
• Substitute \(a = \frac{1}{20}\) and \(x = 10\) into the equation.
• The calculation \( \frac{1}{20} \times 10 \) yields \( \frac{1}{2} \), thus confirming our solution.

By using substitution, we ensure that the values we found work correctly within the system of the equation. It's a straightforward method for validating solutions and is widely used in algebra to simplify complex problems into manageable calculations.