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Chapter 2: Problem 29

Solve and check. Label any contradictions or identities. $$4+\frac{7}{2} x=-10$$

Short Answer

Expert verified
The solution is \( x = -4 \). It satisfies the original equation with no contradictions or identities.

Step by step solution

01

- Simplify the Equation

First, isolate the term containing the variable by subtracting 4 from both sides of the equation:\[ 4 + \frac{7}{2} x - 4 = -10 - 4 \]Simplifying both sides, we get:\[ \frac{7}{2} x = -14 \]
02

- Solve for the Variable

Next, solve for \( x \) by multiplying both sides of the equation by \( \frac{2}{7} \):\[ \frac{2}{7} \times \frac{7}{2} x = -14 \times \frac{2}{7} \]This simplifies to:\[ x = -4 \]
03

- Check the Solution

To verify the solution, substitute \( x = -4 \) back into the original equation:\[ 4 + \frac{7}{2}(-4) = -10 \]Simplify the expression inside the equation:\[ 4 + \frac{7 \times -4}{2} = 4 + (-14) = -10 \]Both sides of the equation are equal, so the solution is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

simplifying equations
Simplifying equations is a crucial step when solving any algebraic problem. To simplify means to make the equation easier to work with. This often involves removing fractions, combining like terms, or isolating variable terms.

In our exercise, we first focus on isolating the term with the variable. For the equation:\[4 + \frac{7}{2} x = -10\]we subtract 4 from both sides to isolate the term containing the variable:
\[4 + \frac{7}{2} x - 4 = -10 - 4\]which simplifies to:
\[\frac{7}{2} x = -14\]

By doing this, we've removed the constant term (4) that doesn鈥檛 contain the variable. The equation is now simpler and easier to solve.
solving for a variable
After simplifying an equation, the next step is solving for the variable. This means manipulating the equation to find the value of the unknown that makes the equation true.

In our example after simplification, we have:\[\frac{7}{2} x = -14\]To solve for \(x\), we need to isolate it completely. Here, we multiply both sides by the reciprocal of \(\frac{7}{2}\), which is \(\frac{2}{7}\):
\[\frac{2}{7} \times \frac{7}{2} x = -14 \times \frac{2}{7}\]
This simplifies to:
\[x = -4\]

By isolating \(x\), we have found the value that satisfies the equation.
verification of solutions
Verification of solutions is a critical step in solving equations. It ensures that the solution obtained is correct and satisfies the original equation.

To verify our solution, we substitute \(x = -4\) back into the original equation:\[4 + \frac{7}{2}(-4) = -10\]By simplifying the expression inside, we get:
\[4 + \frac{7 \times -4}{2} = 4 + (-14) = -10\]

Since both sides of the original equation are equal when \(x = -4\), our solution is verified as correct. This step not only confirms the accuracy of the solution but also gives confidence in the method used.

Always remember to verify your solutions to catch any possible mistakes made during the process.

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