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Chapter 1: Problem 1

Determine whether the given value is a solution of the equation. \(4 x+7=9 x-3\) (a) \(x=-2\) (b) \(x=2\)

Short Answer

Expert verified
(a) No, (b) Yes.

Step by step solution

01

Substitute and Simplify for (a)

First, substitute the given value \(x = -2\) into the equation \(4x + 7 = 9x - 3\). Calculate each side of the equation separately to see if both sides are equal.\[\text{Left side: } 4(-2) + 7 = -8 + 7 = -1\] \[\text{Right side: } 9(-2) - 3 = -18 - 3 = -21\]Since \(-1 eq -21\), the given value \(x = -2\) is not a solution.
02

Substitute and Simplify for (b)

Next, substitute the given value \(x = 2\) into the equation \(4x + 7 = 9x - 3\). Calculate each side of the equation separately to verify equality.\[\text{Left side: } 4(2) + 7 = 8 + 7 = 15\] \[\text{Right side: } 9(2) - 3 = 18 - 3 = 15\]Since \(15 = 15\), the given value \(x = 2\) is a solution.

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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 valuable technique for solving equations. It involves replacing a variable with its given value to see if the equation holds true. In this context, the method is used to test whether certain values satisfy a given equation.
  • First, choose the value you are testing, for example, here we consider two possibilities: \( x = -2 \) and \( x = 2 \).
  • Next, substitute this value in place of \( x \) in the equation \( 4x + 7 = 9x - 3 \).
  • Simplify both sides of the equation independently to determine if they are equal.
Substitution provides a straightforward way to check individual solutions without solving the equation in a more complex manner.
By substituting into the equation, we directly verify the equality condition required for the values to be true solutions.
Linear Equations
Linear equations are a fundamental element in algebra. They are called 'linear' because they form a straight line when graphed on a coordinate plane. A typical linear equation looks like \( ax + b = cx + d \), where \( a \), \( b \), \( c \), and \( d \) are constants.
  • Both sides of a linear equation can contain constants, variables, and coefficients (numbers that multiply the variable).
  • The main goal is to solve for the variable, finding its value that makes both sides of the equation equal.
  • In our exercise, the equation \( 4x + 7 = 9x - 3 \) requires balancing both sides by finding an \( x \) value that makes them equal.
Linear equations are simple yet powerful tools used in various aspects of mathematics and real-world problem-solving. They help us understand relationships between variables with clarity and precision.
Solution Verification
Solution verification is the process of confirming whether a proposed solution to an equation is actually valid. It's an essential step to ensure that our calculations and logic are correct.
  • After substituting the test value into the equation, calculate both the left and right sides to see if they match.
  • If the results are equal, the test value is indeed a solution to the equation.
  • If not, then the value does not satisfy the equation, and is not a valid solution.
In the exercise provided, we verified two potential solutions: \( x = -2 \) and \( x = 2 \).
  • For \( x = -2 \), after substitution, the two sides were not equal, indicating it鈥檚 not a solution.
  • For \( x = 2 \), the matched sides confirmed it as a valid solution.
Checking your work is a good mathematical habit and prevents simple arithmetic errors from leading to incorrect conclusions.

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Most popular questions from this chapter

Factor the expression completely. $$9 x^{2}-36 x-45$$

Plot the points \(P(0,3)\) \(Q(2,2),\) and \(R(5,3)\) on a coordinate plane. Where should the point \(S\) be located so that the figure \(P Q R S\) is a parallelogram? Write a brief description of the steps you took and your reasons for taking them.

Factor out the common factor. $$y(y-6)+9(y-6)$$

Factor the expression completely. $$y^{3}-3 y^{2}-4 y+12$$

Use a Special Factoring Formula to factor the expression. $$16 z^{2}-24 z+9$$
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