Problem 28
Question
Simplify each side of the following equations before applying the addition property. $$x+6-2=5-12$$
Step-by-Step Solution
Verified Answer
The solution is \(x = -11\).
1Step 1: Simplify Left Side
The left side of the equation is given as \(x + 6 - 2\). We can simplify this by performing the subtraction: \(6 - 2 = 4\). So, the simplified left side is \(x + 4\).
2Step 2: Simplify Right Side
The right side of the equation is given as \(5 - 12\). We can simplify this by performing the subtraction: \(5 - 12 = -7\). So, the simplified right side is \(-7\).
3Step 3: Rewrite the Equation with Simplified Sides
After simplifying both sides, the equation becomes \(x + 4 = -7\).
4Step 4: Apply the Subtraction Property
To isolate \(x\), subtract 4 from both sides of the equation: \(x + 4 - 4 = -7 - 4\).
5Step 5: Solve for x
Simplify both sides: \(x = -11\). The value of \(x\) that satisfies the equation is \(-11\).
Key Concepts
Addition PropertySubtraction PropertySolving Linear EquationsPrealgebra Concepts
Addition Property
The addition property of equality is a fundamental concept used in solving equations. It states that if you add the same number to both sides of an equation, the equality remains unchanged. This principle is essential in keeping the balance of an equation while simplifying or solving it.
Imagine a seesaw with equal weights on both sides. If you add the same weight to both sides, it stays balanced. This analogy holds true for equations as well. In the context of solving equations, this property is frequently used to help isolate variables or to move terms across the equation.
For example, if we have the equation \(x + 2 = 5\), and we want to solve for \(x\), we could use the addition property by adding \(-2\) (which is essentially applying the subtraction name but carries the idea of adding a negative) to both sides, resulting in \(x + 2 - 2 = 5 - 2\). This would simplify to \(x = 3\). Thus, using the addition property helps maintain the balance while rewriting equations.
Imagine a seesaw with equal weights on both sides. If you add the same weight to both sides, it stays balanced. This analogy holds true for equations as well. In the context of solving equations, this property is frequently used to help isolate variables or to move terms across the equation.
For example, if we have the equation \(x + 2 = 5\), and we want to solve for \(x\), we could use the addition property by adding \(-2\) (which is essentially applying the subtraction name but carries the idea of adding a negative) to both sides, resulting in \(x + 2 - 2 = 5 - 2\). This would simplify to \(x = 3\). Thus, using the addition property helps maintain the balance while rewriting equations.
Subtraction Property
The subtraction property of equality allows you to subtract the same amount from each side of an equation without affecting the balance of the equation. It's a crucial method for isolating variables.
Think of an equation as a balanced scale. Removing the same weight from either side keeps that scale balanced. For equations, this principle helps in moving or removing terms across the equality.
Conside the equation \(x + 4 = -7\) as seen in our given problem. Here, the goal is to solve for \(x\). We use the subtraction property by subtracting 4 from both sides, simplifying it to \(x = -7 - 4\), which further simplifies to \(x = -11\). The subtraction property is indispensable in solving equations like this as it provides a straightforward method to simplify problems.
Think of an equation as a balanced scale. Removing the same weight from either side keeps that scale balanced. For equations, this principle helps in moving or removing terms across the equality.
Conside the equation \(x + 4 = -7\) as seen in our given problem. Here, the goal is to solve for \(x\). We use the subtraction property by subtracting 4 from both sides, simplifying it to \(x = -7 - 4\), which further simplifies to \(x = -11\). The subtraction property is indispensable in solving equations like this as it provides a straightforward method to simplify problems.
Solving Linear Equations
Solving linear equations is the process of finding the value of the variable that makes the equation true. These types of equations involve variables raised to the power of one and make up a fundamental aspect of algebra.
When solving linear equations, you often utilize properties of equality like addition and subtraction, as mentioned above, to simplify and eventually solve the equation.
The goal is to isolate the variable on one side of the equation. Consider the equation given in our exercise: \(x + 4 = -7\). Our task is to find the value of \(x\) that satisfies the equation. We achieve this by using the subtraction property to isolate \(x\) and simplify both sides, as follows: subtract 4 from both sides to yield \(x = -11\). Linear equations are convenient to solve due to their predictable nature and foundational algebraic techniques that apply across a variety of problems.
When solving linear equations, you often utilize properties of equality like addition and subtraction, as mentioned above, to simplify and eventually solve the equation.
The goal is to isolate the variable on one side of the equation. Consider the equation given in our exercise: \(x + 4 = -7\). Our task is to find the value of \(x\) that satisfies the equation. We achieve this by using the subtraction property to isolate \(x\) and simplify both sides, as follows: subtract 4 from both sides to yield \(x = -11\). Linear equations are convenient to solve due to their predictable nature and foundational algebraic techniques that apply across a variety of problems.
Prealgebra Concepts
Prealgebra serves as the initial building block for all algebraic understanding. It includes fundamental concepts and operations such as addition, subtraction, basic equation solving, and number properties.
These skills are crucial as they lay the groundwork for more advanced mathematics. Understanding how to manipulate numbers and variables through these operations enables students to tackle more complex problems confidently.
In our exercise, the simplification of expressions like \(x + 6 - 2\) and \(5 - 12\), as initial steps, demonstrates prealgebra concepts in action. Simplifying expressions is also a keen part of prealgebra that paves the way to more intricate mathematical operations. By mastering these foundational skills, students create a strong mathematical base to build upon as they advance in their studies.
These skills are crucial as they lay the groundwork for more advanced mathematics. Understanding how to manipulate numbers and variables through these operations enables students to tackle more complex problems confidently.
In our exercise, the simplification of expressions like \(x + 6 - 2\) and \(5 - 12\), as initial steps, demonstrates prealgebra concepts in action. Simplifying expressions is also a keen part of prealgebra that paves the way to more intricate mathematical operations. By mastering these foundational skills, students create a strong mathematical base to build upon as they advance in their studies.
Other exercises in this chapter
Problem 28
Using the addition property of equality first, solve each of the following equations. $$\frac{1}{2} a+2=-7$$
View solution Problem 28
One angle in a triangle is half the largest angle, but three times the smallest. Find all three angles.
View solution Problem 28
Simplify the following expressions by combining similar terms. In some cases the order of the terms must be rearranged first by using the commutative property.
View solution Problem 28
Solve each equation using the methods shown in this section. $$8 x-10=-4(x+3)+2$$
View solution