Problem 36
Question
Solve each equation using the addition property of equality. Be sure to check your proposed solutions. $$-11=8+x$$
Step-by-Step Solution
Verified Answer
The solution to the equation -11=8+x using the addition property of equality is \( x = -19 \). This solution has been verified and is correct.
1Step 1: Apply the Addition Property of Equality
The purpose is to isolate the variable 'x' on one side of the equation. To do this, subtract 8 from both sides of the equation -11=8+x, to get -11-8 = 8+x-8.
2Step 2: Simplify Both Sides of the Equation
After subtraction, now simplify both sides of the equation to get -19 = x.
3Step 3: Check the Proposed Solution
To ensure that the obtained value is correct, substitute 'x' in the original equation with -19. Doing so will give -11 = 8 - 19. When this equation is simplified, it equals -11 = -11, which verifies that the solution obtained is correct.
Key Concepts
Solving Linear EquationsIsolating VariablesEquation Checking
Solving Linear Equations
When we are faced with a linear equation like \( -11 = 8 + x \), the primary goal is to find the value of the variable that makes the equation true. Solving linear equations can seem daunting at first, but by following a systematic approach, we can make it much simpler.
The first step is to look at the equation and identify the variable we need to solve for, in this case, \( x \). Then we proceed to isolate this variable on one side of the equation. This process usually involves applying operations equally to both sides of the equation to maintain the balance, which is known as the addition property of equality.
It’s like a scale; whatever you do to one side, you must do to the other to keep it balanced. For our example, we subtract \( 8 \) from both sides because this is the inverse operation of what's been applied to \( x \) in the equation. Through this process, the variable \( x \) becomes isolated, making \( x = -19 \).
Solving linear equations is a fundamental skill in algebra and it's important because it allows us to find unknown values that are critical in various applications, from simple daily tasks to complex scientific calculations.
The first step is to look at the equation and identify the variable we need to solve for, in this case, \( x \). Then we proceed to isolate this variable on one side of the equation. This process usually involves applying operations equally to both sides of the equation to maintain the balance, which is known as the addition property of equality.
It’s like a scale; whatever you do to one side, you must do to the other to keep it balanced. For our example, we subtract \( 8 \) from both sides because this is the inverse operation of what's been applied to \( x \) in the equation. Through this process, the variable \( x \) becomes isolated, making \( x = -19 \).
Solving linear equations is a fundamental skill in algebra and it's important because it allows us to find unknown values that are critical in various applications, from simple daily tasks to complex scientific calculations.
Isolating Variables
Isolating the variable in an equation is a key step in finding its solution. This means that we want to get the variable, \( x \) in this case, by itself on one side of the equation. To isolate the variable, we use inverse operations to cancel out other numbers or variables that are on the same side as the variable we're trying to solve for.
For example, in the equation \( -11 = 8 + x \), \( x \) is accompanied by \( 8 \). To isolate \( x \), we subtract \( 8 \) from both sides to get rid of it. Remember, whatever operation you do to one side, you must do to the other. This gives us the equation \( -11 - 8 = x \) which simplifies to \( -19 = x \).
It’s crucial to perform the operations correctly and in the right order. Always deal with addition and subtraction first before moving onto any multiplication or division, unless the equation calls for a different operation to isolate the variable first. Understanding how to isolate variables is a vital skill for solving not just linear equations, but many other types of algebraic equations as well.
For example, in the equation \( -11 = 8 + x \), \( x \) is accompanied by \( 8 \). To isolate \( x \), we subtract \( 8 \) from both sides to get rid of it. Remember, whatever operation you do to one side, you must do to the other. This gives us the equation \( -11 - 8 = x \) which simplifies to \( -19 = x \).
It’s crucial to perform the operations correctly and in the right order. Always deal with addition and subtraction first before moving onto any multiplication or division, unless the equation calls for a different operation to isolate the variable first. Understanding how to isolate variables is a vital skill for solving not just linear equations, but many other types of algebraic equations as well.
Equation Checking
After solving the equation and finding the value of the variable, it's important not to assume the work is done. Equation checking is the final, yet crucial, step in the problem-solving process. This step ensures that our proposed solution actually works and satisfies the original equation.
To check the solution \( x = -19 \) for our equation \( -11 = 8 + x \), we substitute \( -19 \) back into the equation in place of \( x \). This gives us \( -11 = 8 - 19 \), and simplifying the right side results in \( -11 = -11 \). This equality verifies that our solution is correct.
Equation checking prevents errors that might have arisen during the process of isolating the variable and solving the equation. Always include this step before considering a problem complete, as it is a demonstration of mathematical proof that supports the accuracy of your solution, and teaches you to verify your work systematically.
To check the solution \( x = -19 \) for our equation \( -11 = 8 + x \), we substitute \( -19 \) back into the equation in place of \( x \). This gives us \( -11 = 8 - 19 \), and simplifying the right side results in \( -11 = -11 \). This equality verifies that our solution is correct.
Equation checking prevents errors that might have arisen during the process of isolating the variable and solving the equation. Always include this step before considering a problem complete, as it is a demonstration of mathematical proof that supports the accuracy of your solution, and teaches you to verify your work systematically.
Other exercises in this chapter
Problem 35
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