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Chapter 15: Problem 91

Determine whether the given number is a solution of the given equation. $$ -3 ; 4-x=1 $$

Short Answer

Expert verified
-3 is not a solution.

Step by step solution

01

- Substitute the number into the equation

To determine whether the given number -3 is a solution of the equation, substitute -3 for the variable x in the equation. The equation is 4 - x = 1. By substituting -3 for x, the equation becomes 4 - (-3) = 1.
02

- Simplify the left side

Simplify the left side of the equation 4 - (-3) = 1. When you subtract a negative number, it is the same as adding its positive counterpart. Thus, 4 - (-3) simplifies to 4 + 3, which equals 7. This gives us: 7 = 1.
03

- Evaluate the equation

Compare the simplified left side to the right side of the equation. We have 7 on the left and 1 on the right, which are not equal. Therefore, -3 is not a solution to the equation 4 - x = 1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

substitution method in algebra
The substitution method in algebra is a powerful technique used to determine if a given number satisfies an equation. This involves replacing a variable in the equation with the number in question. For example, consider the equation 4 - x = 1. To check if -3 is a solution:

- Substitute -3 for x.
- Replace x with -3, transforming the equation into 4 - (-3) = 1.
This method lets you see if the equation holds true with the substituted value, thus confirming if it is a valid solution.
negative numbers
Handling negative numbers correctly is crucial in algebra. When substituting and simplifying equations, it鈥檚 vital to remember the rules for operations involving negatives:

- Subtracting a negative number is the same as adding its positive counterpart.
- For example, in our equation after substituting -3 for x, we get 4 - (-3). Here, subtracting -3 is equivalent to adding 3, changing the expression to 4 + 3.
This simplification is key to understanding how negative values interact in equations.
equation simplification
Simplifying equations helps in understanding their true form and checking correctness. After substitution in our example:

- Start with the modified equation 4 - (-3) = 1.
- Recognize that 4 - (-3) simplifies to 4 + 3.
- Simplify further to get 7 = 1.
Simplification makes it clear whether both sides of the equation are balanced, thereby verifying whether the initial value was a valid solution.
variable evaluation
Evaluating variables involves comparing simplified expressions to see if both sides of an equation are equal. In our example:

- After simplification, we get 7 = 1.
- Since 7 is not equal to 1, we conclude that -3 is not a solution to the original equation 4 - x = 1.
Proper evaluation ensures that each variable substitution is accurately tested, confirming or denying it as a viable solution.

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Most popular questions from this chapter

Solve each system using the substitution method. $$ \begin{aligned} &3 s-4 t=14\\\ &t=8-5 s \end{aligned} $$

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Simplify. $$ 2+8 \div(2 \cdot 2) $$

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