Problem 9 - Precalculus: Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

(a) Not a solution, (b) Is a solution

Step 1: Substitute x = -2 into the equation

First, we substitute \(x = -2\) into the equation \(4x + 7 = 9x - 3\). This gives us \(4(-2) + 7 = 9(-2) - 3\). Simplifying both sides, we calculate \(-8 + 7 = -18 - 3\). Simplifying further gives us \(-1 eq -21\). Hence, \(x = -2\) is not a solution.

Step 2: Substitute x = 2 into the equation

Next, we substitute \(x = 2\) into the equation \(4x + 7 = 9x - 3\). We get \(4(2) + 7 = 9(2) - 3\). Simplifying these expressions, we get \(8 + 7 = 18 - 3\). This simplifies to \(15 = 15\). Therefore, \(x = 2\) is a solution.

Key Concepts

Substitution Method

The substitution method is a powerful tool when solving algebraic equations. It involves replacing a variable in the equation with a given value to see if the equation holds true. In simpler terms, you "substitute" the value you're checking into every instance of the variable:

• Find the variable: Typically represented by letters like \(x\) or \(y\).
• Replace it: Substitute the variable with the given value.
• Simplify: Calculate each side of the equation separately to see if the statement is logically correct.

In this exercise, the given values were substituted into the equation \(4x + 7 = 9x - 3\). This shows how each side of the equation is evaluated to determine if they are equal. If both sides equal each other after substitution, the value is a solution.

Linear Equations

Linear equations are a fundamental part of algebra. They represent equations of straight lines, and they usually look like \(ax + b = cx + d\). Here's why linear equations are simple yet important:

• Linear: The term "linear" indicates that the variable is to the first power (no squares, cubes, etc.).
• Balance: Whatever you do to one side of the equation, you must do to the other, maintaining balance.
• Straight Line: Graphically, they represent straight lines on a coordinate plane.

In the example \(4x + 7 = 9x - 3\), both sides of the equation involve the variable \(x\) and constants, making it linear. Solving such equations typically involves isolating \(x\) but sometimes, like in this task, just checking substitute values.

Solution Verification

Verifying solutions in equations ensures accuracy. Verification involves two essential steps:

• Substitution: Plug the proposed solution back into the original equation.
• Check Equality: Simplify to confirm if the two sides of the equation produce an identical result.

In the original problem, to verify whether \(x = -2\) or \(x = 2\) was the solution, each value was tested separately. For \(x = 2\), when substituted, both sides simplified to 15, confirming it as a solution. Meanwhile, \(x = -2\) yielded different results on either side, proving it wasn't a valid solution. Verification is crucial in math to ensure solutions are correct and reliable.