Problem 6 - Elementary Algebra

Question

Solve the equations and inequalities for the following problems. $$ 5 b-8=7 b+12 $$

Step-by-Step Solution

Answer

Answer: The value of b in the given equation is -10.

Step 1: Write down the given equation

The given equation is: $$ 5b - 8 = 7b + 12 $$

Step 2: Move the terms containing the variable (b) to one side of the equation

Now, we'll move the term containing b from the right side of the equation to the left side. To do this, we'll subtract \(7b\) from both sides of the equation: $$ 5b - 7b - 8 = 12 $$

Step 3: Simplify the left side of the equation

Now, we'll simplify the left side of the equation by combining the like terms: $$ -2b - 8 = 12 $$

Step 4: Move the constant term to the right side of the equation

To isolate b, we'll move the constant term (-8) to the right side of the equation. This can be done by adding 8 to both sides: $$ -2b = 12 + 8 $$

Step 5: Simplify the right side of the equation

Now, we'll simplify the right side of the equation by adding the constant terms: $$ -2b = 20 $$

Step 6: Solve for the variable (b)

Finally, we'll solve for b by dividing both sides of the equation by -2: $$ b = \frac{20}{-2} $$

Step 7: Calculate the value of the variable (b)

Now, we'll find the value for b: $$ b = -10 $$ So the solution of the equation is \(b = -10\).

Key Concepts

Algebraic Manipulation

Algebraic manipulation is a core technique used in solving equations and inequalities. It's essentially the process of rearranging terms in an equation to isolate a specific variable. By performing operations such as addition, subtraction, multiplication, and division on both sides of an equation, we can simplify and solve it efficiently. This process requires careful attention to maintaining balance between both sides of the equation.

When manipulating equations, there are a few key steps to remember:

• Always perform the same operation on both sides of the equation.
• Combine like terms whenever possible to simplify the equation.
• Keep the equation balanced to ensure the solution remains valid.

For instance, in the example given, we subtracted \(7b\) from both sides to move the variable terms together, followed by combining like terms and simplifying to isolate the desired variable.

Linear Equations

Linear equations are equations that make a straight line when graphed. They typically take the form \(ax + b = c\), where \(x\) is the variable, and \(a\), \(b\), and \(c\) are constants. These equations are known for their simplicity and are often the first type of equation taught in algebra. Solving them requires a straightforward process, often involving basic algebraic techniques.

In the given example, the equation \(5b - 8 = 7b + 12\) is a linear equation with the variable \(b\). Because it only involves the first power of \(b\), it remains a simple case of balancing and manipulation. Linear equations like this are foundational and useful, as they illustrate basic principles of algebra and prepare students for more complex mathematical concepts.

Isolation of Variables

Isolation of variables is a crucial step in solving equations, especially when dealing with linear equations. It involves reordering an equation so that a particular variable stands alone on one side. By doing this, you can determine the variable's value independently, effectively solving the equation.

To isolate a variable, typically:

• Move all terms with the variable of interest to one side of the equation.
• Move constant terms to the opposite side.
• Simplify the equation by performing the appropriate operations to clear out coefficients or any unwanted terms associated with the variable.

In our example, we moved \(-8\) to the other side by adding \(8\) and then solved for \(b\) by dividing both sides by \(-2\), which left \(b\) isolated, confirming that \(b = -10\). Mastery of isolating variables is fundamental in solving all kinds of equations.