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Chapter 6: Problem 58

Solve each equation. $$\frac{3 a}{2}+\frac{a}{3}=-22$$

Short Answer

Expert verified
The solution to the equation is \(a = -12.\)

Step by step solution

01

Eliminate fractions

Multiply the entire equation by 6 (the least common multiple of 2 and 3) to eliminate the fractions: \[6 \times (\frac{3a}{2}+\frac{a}{3}) = 6 \times -22.\] After multiplying, the equation simplifies to \[9a + 2a = -132.\]
02

Combine like terms

Combine terms that involve the variable \(a\) on the left side of the equation: \[9a + 2a = 11a.\] So, the equation is now \[11a = -132\].
03

Solve for \(a\)

To find the value of \(a\), divide both sides of the equation by 11: \(a = \frac{-132}{11}.\)

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Key Concepts

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

Fractions in Equations
Dealing with fractions in equations can often make solving seem tricky, but there's a method to simplify things! In algebra, fractions can obscure the relationship between variables and constants. To streamline the process, we look for ways to eliminate fractions from our equations.

The key is to find the Least Common Multiple (LCM) of the denominators in the equation. This LCM helps transform the equation into a simpler, more manageable form.

  • Identify the denominators: In the problem, these are 2 and 3.
  • Calculate the LCM: Here, it鈥檚 6.
  • Multiply every part of the equation by this LCM.
This action clears the fractions, allowing us to directly solve for the variable. This approach simplifies the equation, making the arithmetic straightforward and setting the stage for solving the equation.
Combining Like Terms
Once fractions are eliminated, it's time to combine like terms. Like terms are the parts of an equation that have the same variable raised to the same power. Simplifying by combining these terms reduces the complexity of the equation and brings us closer to finding the solution.

In our example, after eliminating fractions, the equation becomes:

- \(9a + 2a = -132\)

Combining like terms involves:
  • Adding or subtracting coefficients of the same variable.
  • Ensuring that all like terms are consolidated on one side of the equation.
In this case, we combine the coefficients 9 and 2:

- \(11a = -132\)
This simplified form makes the next steps more straightforward and clearly shows the relationship between the variable and the constants in the equation.
Solving for Variables
With the equation simplified, our next focus is to solve for the variable. This means isolating the variable on one side of the equation to find its value. This process involves basic arithmetic operations to manipulate the equation into a form that shows the variable's value.

In this equation, we need to solve for \(a\):

- \(11a = -132\)
To isolate \(a\), divide both sides by 11:
  • This leaves \(a\) alone on one side of the equation.
  • You perform the division on each side of the equation to maintain equality.
Thus, we find:

- \(a = \frac{-132}{11}\)
This step gives us the solution, showing the value of \(a\) that satisfies the original algebraic equation. Solving for variables is the goal in algebraic equations, revealing the unknowns by systematically applying mathematical operations.

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Most popular questions from this chapter

Solve each equation. Solve the formula \(\frac{1}{r}=\frac{1}{r_{1}}+\frac{1}{r_{2}}\) for \(r_{1}\).

A recipe for spaghetti sauce requires four 16 -ounce bottles of catsup to make two gallons of sauce. How many bottles of catsup are needed to make 10 gallons of sauce?

Write each expression in simplest form. If it is already in simplest form, so indicate. Assume that no denominators are 0. SEE EXAMPLE 10. (OBJECTIVE 4) $$\frac{x-y}{y-x}$$

Write each expression in simplest form. If it is already in simplest form, so indicate. Assume that no denominators are \(0 .\) $$\frac{x z-2 x}{y z-2 y}$$

$$\text { Solve: } x^{-2}+x^{-1}=0$$.
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