Problem 46 - Prealgebra

Question

Solve each equation. Check your solution. $$-3+2 z=-19$$

Step-by-Step Solution

Answer

The solution is \(z = -8\).

Step 1: Isolate the term with the variable

To solve the equation \(-3 + 2z = -19\), we need to isolate terms involving \(z\). Start by eliminating the constant term on the left-hand side by adding 3 to both sides of the equation:\[-3 + 2z + 3 = -19 + 3\] which simplifies to:\[2z = -16\]

Step 2: Solve for the variable

Now that we have isolated the term with \(z\), the equation is \(2z = -16\).Divide both sides by 2 to solve for \(z\):\[\frac{2z}{2} = \frac{-16}{2}\]This simplifies to\(z = -8\).

Step 3: Check the solution

Substitute \(z = -8\) back into the original equation to verify the solution is correct:The original equation is \(-3 + 2z = -19\).Replace \(z\) with \(-8\):\(-3 + 2(-8) = -19\).Calculate:\(-3 - 16 = -19\).Since both sides of the equation are equal, \(z = -8\) is the correct solution.

Key Concepts

Equation Solving

Solving linear equations is the process of finding the value of a variable that makes the equation true. In this process, you start with an equation that has a variable on one or both sides. The goal is to manipulate the equation so that you can isolate the variable and determine its value.
Linear equations are the simplest kinds of equations, containing terms that add to or subtract from linear expressions. For example, the equation \(-3 + 2z = -19\) can be understood as a balance scale. Each side of the equation must be equal. Your mission is to find out what number can be put in place of the variable, here "\(z\)", so that both sides remain equal.

• **Keep equations balanced:** Whatever you do to one side of the equation, you must also do to the other to maintain equality.
• **Isolate the variable:** This means rearranging the equation so that the variable is alone on one side of the equation.
• **Substitute to verify:** Substitute the solution back into the original equation to ensure that both sides are equal, confirming the solution's validity.

Prealgebra

Prealgebra serves as the foundation for all future algebraic concepts. It includes fundamental operations on numbers and introduces students to the idea of solving equations.
Prealgebra prepares students for Algebra 1 by teaching them to work with integers, fractions, and decimals, as well as involving them in solving simple linear equations like \(-3 + 2z = -19\). Comprehending prealgebra involves mastering operations such as addition, subtraction, multiplication, and division, which are essential when dealing with equations.

• **Understanding integers:** Prealgebra often uses positive and negative numbers, such as in the exercise given.
• **Basic operations:** Addition and subtraction are keys to moving terms around, while division and multiplication help with isolating variables.
• **Maintaining equilibrium:** Ensuring that the operations maintain balance on both sides of the equation is a primary skill taught in prealgebra.

Variable Isolation

Variable isolation is a crucial step in solving algebraic equations. The term "isolating the variable" means rearranging the equation so that the variable stands alone on one side, making it easier to identify its value.
In the equation \(-3 + 2z = -19\), the aim is to perform arithmetic operations that eliminate all other numbers from the side of the equation with the variable \(z\), leaving \(z\) by itself.

• **Move constants first:** Add or subtract constants from both sides to move them away from the variable side.
• **Divide or multiply:** Once you have the variable term isolated, divide or multiply to solve for the variable. For example, divide both sides by 2 to solve \(2z = -16\), resulting in \(z = -8\).
• **Verify the results:** After isolating and finding the variable's value, plug it back into the original equation to confirm the accuracy of your solution.