Problem 89 - Introductory Algebra for College Students

Question

Is 4 a solution of \(3 x-14=-2 x+6 ?\)

Step-by-Step Solution

Answer

Yes, 4 is a solution of the equation \(3x - 14 = -2x + 6\)

Step 1: State the equation

Here is the linear equation provided: \(3x - 14 = -2x +6 \)

Step 2: Substitute the number 4 into the equation

Replacing 'x' with 4 in the equation gives: \(3*4 - 14 = -2*4 +6 \). We now do the arithmetic: \(12 - 14 = -8 + 6 \) which simplifies to \(-2 = -2\). As both sides are the same, this statement is true.

Step 3: Conclusion

Since substituting 4 for 'x' resulted in a true statement, we can conclude that 4 is a solution of the given equation.

Key Concepts

Linear Equation

A linear equation is an equation involving only the first power of a variable. It is the simplest form of an equation that can be formed with a mathematical expression. This type of equation is written in the form:

• \( ax + b = 0 \)

where \( a \) and \( b \) are constants, and \( x \) is the variable to solve for. In our exercise, the linear equation given was \( 3x - 14 = -2x + 6 \). This equation describes a line when plotted on a graph.
Linear equations are important because they are foundational in algebra and are used for finding relationships between variables. The solutions to linear equations are the values of \( x \) that make the equation true. Understanding how to manipulate and solve them is crucial for solving more complex algebraic problems.

Solution of an Equation

The solution of an equation is a value or set of values that satisfy the equation, making the statement true. When we say a number is a solution to an equation, it means that when you replace the variable in the equation with this number, both sides of the equation will be equal.
In the exercise, we were asked whether 4 is a solution of the equation \( 3x - 14 = -2x + 6 \). By substituting 4 for \( x \) and simplifying both sides, we found that the left side equaled the right side, leading to the statement \( -2 = -2 \). This equality indicates that 4 is indeed a solution.
Solutions are significant because they help us understand where certain scenarios meet established conditions. In real-life applications, they can represent anything from financial balances to scientific measurements.

Substitution Method

The substitution method is a widely used technique for solving equations, especially systems of equations. It involves substituting, or replacing, one variable with an equivalent expression from another equation. This reduces the problem to an equation with a single variable, making it easier to solve.
In our exercise, we used the substitution method in a simple form. We replaced \( x \) in the equation \( 3x - 14 = -2x + 6 \) with the number 4 to check if it was a solution. This substitution transformed our equation into a straightforward numeric calculation: \( 12 - 14 = -8 + 6 \), which we simplified and found to be true.

• The substitution method is useful because it simplifies complex equations.
• It helps break down problems into manageable steps.
• It's applicable in both algebra and calculus for solving equations and inequalities.

Understanding substitution empowers you to solve different types of equations effectively, by isolating variables and finding solutions efficiently.