Problem 2 - Algebra A Combined Function

Question

Solve each equation. Check each solution. See Examples 1 through \(6 .\) \(-7 x=-49\)

Step-by-Step Solution

Answer

The solution is \(x = 7\).

Step 1: Isolate the variable

We start by isolating the variable \(x\). Currently, the equation is \(-7x = -49\). To isolate \(x\), we need to divide both sides of the equation by \(-7\). This gives us: \[ x = \frac{-49}{-7} \]

Step 2: Simplify the fraction

Now, simplify the fraction \(\frac{-49}{-7}\). We calculate the division: \[ x = 7 \] because dividing \(-49\) by \(-7\) equals \(7\).

Step 3: Verification

To verify the solution, substitute \(x = 7\) back into the original equation: \(-7(7) = -49\). Calculating the left side: \(-49 = -49\), which is a true statement. Thus, \(x = 7\) is indeed the correct solution.

Key Concepts

Solving Linear Equations

Solving linear equations is a fundamental skill in algebra, which involves finding the value of the variable that makes the equation true. Linear equations are equations of the first degree, which means they have variables raised only to the power of one.

To solve a linear equation, you typically follow these steps:

• Identify the equation and recognize it's a linear equation (like the one in the exercise: \(-7x = -49\)).
• Manipulate the equation to get the variable on one side and the constant on the other.
• Use basic arithmetic operations (addition, subtraction, multiplication, or division) to simplify and solve for the variable.

In our example, the equation \(-7x = -49\) is linear because it involves multiplying a variable (\(x\)) by a constant (-7). The goal here is to isolate \(x\) to determine its value.

Variable Isolation

Variable isolation is the process of rearranging an equation in such a way that the variable we're solving for stands alone on one side. In our example, this means having only \(x\) on one side of the equation.

• The main tool for isolating the variable is using inverse operations to "undo" what's being done to the variable.
• In the equation \(-7x = -49\), \(x\) is being multiplied by \(-7\).
• To isolate \(x\), perform the inverse operation; divide both sides by \(-7\).

This math operation results in:\[ x = \frac{-49}{-7} \]Simplifying gives us \(x = 7\). The division essentially "cancels out" the multiplication by \(-7\), leaving \(x\) by itself. Variable isolation is a key step because it helps us find the exact value of \(x\) or any variable in an equation.

Verifying Solutions

Verifying solutions ensures that the value found for the variable actually satisfies the equation. This is a crucial step because it confirms the correctness of our solution.

Here's how you verify a solution:

• Substitute the solution back into the original equation.
• Perform the operations to check if both sides of the equation are equal.

In our equation \(-7x = -49\), we found \(x = 7\). By substituting \(x = 7\) back into the equation, calculate:\[-7(7) = -49\]After computing, both sides equal \(-49\), confirming that our solution is indeed correct.

Verification is important because it gives you confidence in your answer, ensuring that the original problem's requirements are met.