Problem 138
Question
Is \(-7\) a solution of the equation \(5+a=-9-a ?\)
Step-by-Step Solution
Verified Answer
Yes, -7 is a solution for the equation.
1Step 1: Substitute the provided value into the equation
Replace 'a' in the equation with \(-7\) to get: \(5 + (-7) = -9 - (-7)\).
2Step 2: Simplify the equation
By performing the addition and subtraction operations in the equation, it simplifies to \(-2 = -9 +7\).
3Step 3: Simplify further to check the equality
When the right side of the equation is simplified, it becomes: \(-2 = -2\).
4Step 4: Concluding the result
As the left side of the equation equals the right side, the provided value, \(-7\), is indeed a solution for the equation.
Key Concepts
Substitution MethodSimplifying EquationsChecking Solutions
Substitution Method
The substitution method is a fundamental algebraic technique used to determine if a certain value is a solution to an equation. By substituting the alleged solution directly into the equation, we can assess the equation's validity.
For instance, if you're given an equation such as \(5 + a = -9 - a\) and the number \(-7\) to test, you would start by replacing every instance of 'a' with \(-7\). The paramount step here is accurate substitution ensuring each 'a' is replaced with \(-7\), which changes the equation to \(5 + (-7) = -9 - (-7)\). This must be done carefully because any mistake at this stage can lead to an incorrect evaluation of the solution's validity.
For instance, if you're given an equation such as \(5 + a = -9 - a\) and the number \(-7\) to test, you would start by replacing every instance of 'a' with \(-7\). The paramount step here is accurate substitution ensuring each 'a' is replaced with \(-7\), which changes the equation to \(5 + (-7) = -9 - (-7)\). This must be done carefully because any mistake at this stage can lead to an incorrect evaluation of the solution's validity.
Simplifying Equations
After substituting values, simplifying the equation is crucial for solving algebra problems. Simplification involves combining like terms and performing basic arithmetic operations: addition, subtraction, multiplication, and division.
In the example \(5 + (-7) = -9 - (-7)\), we simplify each side of the equation separately. The left side becomes \(5 - 7\), which simplifies to \(-2\). The right side involves the subtraction of a negative number, which is equivalent to addition, so \(-9 + 7\), also simplifying to \(-2\).
Simplification not only makes the equation easier to read but also facilitates the final steps of solving or checking the solution.
In the example \(5 + (-7) = -9 - (-7)\), we simplify each side of the equation separately. The left side becomes \(5 - 7\), which simplifies to \(-2\). The right side involves the subtraction of a negative number, which is equivalent to addition, so \(-9 + 7\), also simplifying to \(-2\).
Simplification not only makes the equation easier to read but also facilitates the final steps of solving or checking the solution.
Checking Solutions
Once the equation is simplified, double-checking the solution is a good practice to confirm its correctness. This involves verifying if the simplified left-hand side (LHS) and right-hand side (RHS) of the equation are equal.
In the given exercise, after simplification, we have \(-2 = -2\). The LHS equals the RHS, indicating that the value we substituted for 'a' does make the original equation true. Therefore, \(-7\) is indeed a solution to the equation \(5 + a = -9 - a\). It is vital to remember that the equality must hold true for a value to be considered a solution—this is the essence of checking solutions in algebra.
In the given exercise, after simplification, we have \(-2 = -2\). The LHS equals the RHS, indicating that the value we substituted for 'a' does make the original equation true. Therefore, \(-7\) is indeed a solution to the equation \(5 + a = -9 - a\). It is vital to remember that the equality must hold true for a value to be considered a solution—this is the essence of checking solutions in algebra.
Other exercises in this chapter
Problem 137
a. Name two numbers that are 4 units from 2 on the number line. b. Name two numbers that are 5 units from 3 on the number line.
View solution Problem 137
The boiling point of nitrogen is \(-195.8^{\circ} \mathrm{C},\) and the melting point is \(-209.86^{\circ} \mathrm{C} .\) Find the difference between the boilin
View solution Problem 138
Point \(A\) is a point on the number line halfway between \(-9\) and \(3 .\) Point \(B\) is a point halfway between \(A\) and the graph of 1 on the number line.
View solution Problem 138
The boiling point of oxygen is \(-182.962^{\circ} \mathrm{C}\). Oxygen's melting point is \(-218.4^{\circ} \mathrm{C} .\) What is the difference between the boi
View solution