Problem 29 - Introductory Algebra for College Students

Question

Solve each equation. Be sure to check your proposed solution by substituting it for the variable in the original equation. $$10(z+4)-4(z-2)=3(z-1)+2(z-3)$$

Step-by-Step Solution

Answer

The solution to the equation \(10(z+4)-4(z-2)=3(z-1)+2(z-3)\) is \(z = -57\).

Step 1: Expand the Expressions

Start by expanding out the brackets on each side of the equation: \(10z + 40 - 4z + 8 = 3z - 3 + 2z - 6\).

Step 2: Simplify Each Side

Simplify each side of the equation by combining like terms. This gives: \(6z + 48 = 5z - 9\).

Step 3: Isolate z on one side

To isolate z on one side, subtract \(5z\) from both sides and also subtract \(48\) from both sides. This gives: \(z = -57\).

Step 4: Verify the Solution

Substitute \(z = -57\) back into the original equation to check the solution. Left side will be: \(10*(-57+4) - 4*(-57-2)\) and right side will be \(3*(-57-1) + 2*(-57-3)\). After calculating both sides you will conclude that both the sides of equations are equal, therefore \(z = -57\) is the correct solution.

Key Concepts

Solving Equations

Solving equations is about finding the value of an unknown variable that makes the equation true. In our example, we are trying to find the value of \( z \) that satisfies the given equation: \[ 10(z+4) - 4(z-2) = 3(z-1) + 2(z-3) \]Start by expanding the expressions on both sides. This means removing the brackets by multiplying them out. Once expanded, simplify each side by combining like terms, which leads us to a simpler equation. The goal is to reduce the equation to a point where the variable, \( z \), can easily be isolated and solved.It's essential to keep the equation balanced, meaning whatever operation (like addition or subtraction) you do to one side, you must also do to the other. This principle maintains the equality, allowing us to solve for the variable without altering the original equation's meaning.

Verifying Solutions

Verifying solutions in algebra involves checking if the solution you obtained is correct by substituting it back into the original equation. For any proposed solution, plug the value back into the equation to see if both sides equal.In our example, after solving and finding \( z = -57 \), substitute \( z \) back into the equation:- Left Side: Substitute and simplify \( 10(-57+4) - 4(-57-2) \).- Right Side: Substitute and simplify \( 3(-57-1) + 2(-57-3) \).Both computations should yield the same number. If they do, it confirms that the solution is correct. If not, re-evaluate your math steps. Verifying ensures accuracy in your final answer and reinforces your understanding of solving equations.

Like Terms

Like terms are terms in an algebraic expression that have identical variable parts. They can be combined by simply adding or subtracting the coefficients.For example, in the expression \( 10z + 40 - 4z + 8 \), the like terms are \( 10z \) and \( -4z \), which combine to form \( 6z \). Another set of like terms here would be the constant numbers, \( 40 \) and \( 8 \), which add up to \( 48 \).Combining like terms simplifies an expression and makes complex equations easier to work with. This simplification step is crucial in solving algebraic equations as it leads to a clearer path to isolating and solving for the variable.

Isolation of Variables

Isolation of variables is a method for solving equations where you simplify the equation to get the unknown variable by itself on one side of the equation. This is accomplished through inverse operations, like adding or subtracting the same number from both sides.In the equation \( 6z + 48 = 5z - 9 \), we isolate \( z \) by:- Subtracting \( 5z \) from both sides to eliminate \( z \) from the right.- Then, subtract \( 48 \) from both sides to move constants to the right.These steps provide us:\[ z = -57 \]This method is fundamental as it allows you to find the exact point where the variable stands alone, leading directly to your solution. Once isolated, you have a straightforward answer, indicating the calculated solution.