Problem 48
Question
Determine whether the given number is a solution of the equation. $$5 z=30 ; 8$$
Step-by-Step Solution
Verified Answer
No, 8 is not a solution to the equation \(5z = 30\).
1Step 1: Substitute the given number for z
In our equation, \(5z = 30\), replace z with the number given, which is 8. So, our new equation becomes \(5(8) = 30\).
2Step 2: Simplify the equation
Calculate the result of \(5 \times 8\) on the left side of the equation. The left side becomes 40, so our equation is now \(40 = 30\).
3Step 3: Compare the two sides
The left side of the equation 40 is not equal to the right side of the equation which is 30. Therefore, 8 is not a solution to the equation \(5z=30\).
Key Concepts
Solution of an EquationSubstitution MethodEquation Solving Steps
Solution of an Equation
An equation is a mathematical statement that asserts the equality of two expressions.
In the equation \(5z = 30\), \(z\) represents the unknown variable that we want to solve for.
A solution to an equation is simply a value you can substitute into the variable that would make the equation true.
For instance, to find if "8" is a solution, you substitute 8 in place of \(z\) and check if both sides of the equation are equal.
In our exercise, when substituting \(8\) for \(z\), the equation becomes \(5(8) = 30\).
If both sides equate to 30, then 8 is a solution. However, if they don't match, it's not a solution.
In the equation \(5z = 30\), \(z\) represents the unknown variable that we want to solve for.
A solution to an equation is simply a value you can substitute into the variable that would make the equation true.
For instance, to find if "8" is a solution, you substitute 8 in place of \(z\) and check if both sides of the equation are equal.
In our exercise, when substituting \(8\) for \(z\), the equation becomes \(5(8) = 30\).
If both sides equate to 30, then 8 is a solution. However, if they don't match, it's not a solution.
Substitution Method
The substitution method involves replacing the variable with a specific number to test if it makes the equation true.
This technique is useful for checking whether a given number is a solution.
Here’s how you perform substitution:- Take the original equation, \(5z=30\).- Replace the variable \(z\) with the number in question, which is 8.Thus, you update the equation to \(5(8) = 30\).
From here, it is simple arithmetic to check if the number works as a solution.
You multiply and then compare the results with the original equation's right-hand side.
This technique is useful for checking whether a given number is a solution.
Here’s how you perform substitution:- Take the original equation, \(5z=30\).- Replace the variable \(z\) with the number in question, which is 8.Thus, you update the equation to \(5(8) = 30\).
From here, it is simple arithmetic to check if the number works as a solution.
You multiply and then compare the results with the original equation's right-hand side.
Equation Solving Steps
When solving equations, a systematic approach ensures clarity and accuracy.
Following a clear process can help you determine if a number is a solution; here are the steps involved:- **Substitute the Number**: Insert the given number into the equation for the variable. In our case, replace \(z\) with 8, yielding \(5(8) = 30\).- **Simplify the Equation**: Perform any calculations needed. With \(5 \times 8\), you'd calculate the expression on the left, which results in 40.- **Compare Both Sides**: After simplifying, ensure both sides of the equation are equal. Here, \(40\) was not equal to \(30\), indicating that 8 does not satisfy the equation.This step-by-step approach makes it easier to see if a number truly solves the equation by seamlessly combining substitution and simplification.
Following a clear process can help you determine if a number is a solution; here are the steps involved:- **Substitute the Number**: Insert the given number into the equation for the variable. In our case, replace \(z\) with 8, yielding \(5(8) = 30\).- **Simplify the Equation**: Perform any calculations needed. With \(5 \times 8\), you'd calculate the expression on the left, which results in 40.- **Compare Both Sides**: After simplifying, ensure both sides of the equation are equal. Here, \(40\) was not equal to \(30\), indicating that 8 does not satisfy the equation.This step-by-step approach makes it easier to see if a number truly solves the equation by seamlessly combining substitution and simplification.
Other exercises in this chapter
Problem 48
Insert either \(\) in the shaded area between each pair of numbers to make a true statement. $$3 \square -\frac{3}{2}$$
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Simplify each algebraic expression. $$-19 x+10 x$$
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Perform the indicated operation. Where possible, reduce the answer to its lowest terms. $$\frac{1}{8} \cdot \frac{2}{3}$$
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Perform the indicated subtraction. $$3 \pi-(-10 \pi)$$
View solution