Problem 80 - Algebra A Combined Function

Question

Fill in the blanks with numbers of your choice so that each equation has the given solution. Note: Each blank will be replaced with a different number. \(x-\) ________ \(=\) _______; Solution: -10

Step-by-Step Solution

Answer

Fill the blanks with 5 and -15.

Step 1: Understand the Equation Form

The given equation is \(x - a = b\), where each blank should be filled with different numbers \(a\) and \(b\). The solution of this equation, when solved for \(x\), must be -10.

Step 2: Solve the Equation for x

Start by solving the equation \(x - a = b\) for \(x\). Add \(a\) to both sides to obtain: \[ x = a + b \] The given solution for \(x\) is -10, so this equation becomes: \[ a + b = -10 \]

Step 3: Choose Numbers to Fill the Blanks

Select numbers for \(a\) and \(b\) such that \(a + b = -10\) and \(a\) and \(b\) are different. For example, choose \(a = 5\) and \(b = -15\). So, \(5 + (-15) = -10\).

Step 4: Verify the Solution

Verify the chosen numbers by substituting them back into the equation: The equation becomes \(x - 5 = -15\), and solving \(x = -15 + 5 = -10\) gives the solution as -10, which matches the given condition.

Key Concepts

Solving Linear Equations

Linear equations are mathematical statements of equality involving variables and constants. The main goal when working with linear equations is to find the value of the unknown variable. To solve them, operations are performed that maintain the equation's balance
A linear equation can be straightforward like \( x + 3 = 5 \) or more complex. Here it involves basic operations such as addition, subtraction, multiplication, and division. Typically, we simplify or manipulate these operations to isolate the variable on one side of the equation, which helps solve for its value.

• Identify the equation structure: In our problem, the equation is of the form \( x - a = b \). Identifying this helps set a clear path to solving it.
• Perform operations to isolate x:We do this by reversing mathematical operations. Here, we add \( a \) to both sides to get \( x = a + b \).
• Check the balance:After isolation, verify by substituting back to ensure both sides are equal.

Understanding these basic principles is key to solving any linear equation.

Variables in Equations

Variables are symbols that represent unknown values in equations. They form a theory's backbone and allow for general problem-solving approaches.
In algebra, variables like \( x \) become placeholders for these unknown quantities, providing us a way to solve equations by isolating these variables with mathematical operations.Variables can take any value that satisfies the equation. The goal is always to determine this value.

• Symbols for unknowns: A variable like \( x \) represents unknown quantities and gives us something to solve for.
• Assignments of specific values:By solving the equation, we find specific values for variables that satisfy the equation.
• Verification:It's crucial to confirm that the found value, when plugged back into the equation, satisfies the original conditions.

By effectively working with variables, we can explore a wide range of mathematical problems from basic equations to complex systems.

Working with Integers

Integers are whole numbers that include negative numbers, positive numbers, and zero. They are vital in solving algebraic equations, particularly when choosing values to fill the blanks in equations.
When dealing with equations like the one in our problem, choosing integers that satisfy the condition \( a + b = -10 \) is essential.

• Positive and negative integers: These give flexibility in forming sums and differences. For instance, choosing \( a = 5 \) and \( b = -15 \) satisfies the equation \( a + b = -10 \).
• Arithmetic operations:Understanding addition and subtraction with integers allows for creating suitable equations and confirming solutions.
• Verifying outcomes:This involves substitution into the original equation to ensure it's balanced.

Grasping how to manipulate integers empowers problem-solving in algebra, enabling the computation of solutions across a variety of mathematical contexts.