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Chapter 1: Problem 56

Determine whether the given number is a solution of the given equation. $$ 75 ; y+28=93 $$

Short Answer

Expert verified
75 is not a solution.

Step by step solution

01

- Substitute the Given Number into the Equation

Substitute the given number, 75, into the equation in place of y. The equation becomes: \[ 75 + 28 = 93 \]
02

- Simplify the Equation

Add the numbers on the left side of the equation to find the sum: \[ 75 + 28 = 103 \]
03

- Compare Both Sides of the Equation

Compare the simplified left side to the right side of the equation. Check if they are equal: \[ 103 eq 93 \]
04

- Conclusion

Since 103 is not equal to 93, the given number 75 is not a solution to the equation \( y + 28 = 93 \).

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Key Concepts

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

substitution in equations
In algebra, substitution is a powerful tool to determine if a given value is a solution to an equation. It involves replacing a variable with a given number and simplifying the equation to check for equality. For example, consider the equation:

$$ y + 28 = 93 $$

The task is to determine if 75 is a solution. We substitute 75 in place of the variable y. This changes our equation to:

$$ 75 + 28 = 93 $$

This substitution helps us focus only on numerical values, making it easier to simplify and compare the expressions. Substitution is essential in solving algebra problems because it tests whether a given number satisfies the equation.
solving linear equations
Linear equations form the backbone of basic algebra. These equations have variables raised only to the first power and are typically in the format of
$$ ax + b = c $$. To solve such equations, the goal is to isolate the variable on one side. Consider the provided equation:
$$ y + 28 = 93 $$

We substitute 75 in place of y, which simplifies the equation to:
$$ 75 + 28 = 103 $$. Then, we add the numbers on the left side:
$$ 75 + 28 = 103 $$. Simplifying the left side of the equation, we see:

$$ 103 eq 93 $$

Therefore, 75 is not a solution. To solve linear equations efficiently:
  • Isolate the variable
  • Perform arithmetic operations to simplify
  • Check solutions by substituting them back into the original equation
. Linear equations highlight the fundamental algebraic skills required for more complex problem-solving.
comparison of equality in equations
To determine if a number is a solution to an equation, we need to compare both sides of the equation after substitution and simplification. This method verifies whether the left-hand side (LHS) equals the right-hand side (RHS). Consider our equation after substitution and simplification:
$$ 75 + 28 = 103 $$

We then compare this simplified LHS to the original RHS, which is 93:
$$ 103 eq 93 $$

Since 103 is not equal to 93, the comparison shows that 75 is not a valid solution for
$$ y + 28 = 93 $$. When working through algebra problems, always:
  • Simplify both sides as much as possible
  • Directly compare the numerical values on both sides
  • Conclude based on the equality or inequality of the sides
. Comparison of equality ensures the accuracy of your results and confirms whether your solutions are correct.

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