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Chapter 1: Problem 70

Decide whether the given number is a solution of the given equation. \(x-10=-7 ; 3\)

Short Answer

Expert verified
Yes, 3 is a solution.

Step by step solution

01

Understand the Problem

We need to determine if the number \(3\) satisfies the equation \(x - 10 = -7\). This means substituting \(3\) for \(x\) and checking if the left side equals the right side of the equation.
02

Substitute the Value

Substitute \(x = 3\) into the equation: \(3 - 10 = -7\).
03

Simplify the Left Side

Calculate \(3 - 10\). This simplifies to \(-7\).
04

Compare Both Sides

Now, compare the simplified left side, \(-7\), with the right side of the equation, which is also \(-7\). Since both sides are equal, the substitution satisfies the equation.

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Key Concepts

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

Substitution Method
The substitution method is a powerful tool for solving equations. It involves replacing a variable with a number to determine if it satisfies the equation. In our example, the equation given is \( x - 10 = -7 \). To use the substitution method, we substitute the proposed solution, which is the number 3, into the equation in place of \( x \).Here's how it works:
  • Take the given value: here, it's \( x = 3 \).
  • Replace \( x \) with 3 in the equation: Thus, the equation becomes \( 3 - 10 = -7 \).
By substituting the number 3 for \( x \), we can directly check if both sides of the equation are equal after simplification. This method is not only useful for checking solutions but also for finding unknowns in more complex algebraic problems.
Verifying Solutions
Verifying solutions is an essential step in solving equations to ensure that the substituted value satisfies the equation. After substituting \( x = 3 \) into the equation \( x - 10 = -7 \), we simplify the left side:
  • Perform the operation \( 3 - 10 \), which gives \( -7 \).
Next, we compare it to the right side of the equation, which is also \( -7 \). Since both sides are equal, it confirms that the solution is correct.Verifying solutions can help you:
  • Ensure no calculation errors were made.
  • Confirm the correctness of a solution in algebraic problems.
This process is crucial, especially in complex equations, as it reassures that the solution makes sense within the given equation.
Linear Equations
Linear equations are equations that make a straight line when graphed on a coordinate plane. They often take the form \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable. In our problem, the linear equation presented is \( x - 10 = -7 \).Characteristics of linear equations include:
  • They have at most one variable with a degree of one.
  • They produce a straight line graph.
  • Solutions are found by isolating the variable using algebraic operations.
To solve, you might either manipulate the equation to isolate the variable or use methods like substitution to determine solutions. Understanding linear equations is the cornerstone of learning algebra as they form the basis for more complex equations and systems in mathematics.

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Most popular questions from this chapter

Simplify each expression by combining any like terms. $$ 0.4 y-6.7+y-0.3-2.6 y $$

Remove parentheses and simplify each expression. $$ 10-3(2 x+3 y) $$

Write each sentence as an equation or inequality. Use \(x\) to represent any unknown number. 9.1 times a number equals 4

Write each sentence as a mathematical statement. Negative seven is not equal to seven.

Add. See Examples 1 through 12,18, and 19. $$ -6+(-8) $$
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