Problem 44 - Algebra for College Students

Question

Solve each equation, and check the solution. If applicable, tell whether the equation is an identity or a contradiction. \(4[2 x-(3-x)+5]=-(2+7 x)\)

Step-by-Step Solution

Answer

The equation has a contradiction and has no solution.

Step 1 - Simplify the equation

Open the brackets on both sides of the equation. The equation is: a. Expand: a. Left side: a. Simplify: a. Combine terms: b. Right side: 4[2x - (3 - x) + 5] = -(2 + 7x) => 4[2x - 3 + x + 5] = -(2 + 7x)=> 4[3x + 2] = -(2 + 7x)=> 12x + 8 = -2 - 7x

Step 2 - Combine like terms

Combine the like terms on both sides of the equation.12x + 8 = -2 - 7x

Step 3 - Move variables to one side

Add 7x to both sides of the equation to move all the x terms to one side.12x + 7x + 8 = -2 => 19x + 8 = -2

Step 4 - Isolate the variable

Subtract 8 from both sides to isolate the term with the variable.=> 19x + 8 - 8 = -2 - 8 => 19x = -10

Step 5 - Solve for x

Divide both sides by 19 to solve for x.=> x = -10/19

Step 6 - Check the solution

Substitute the value of x back into the original equation to check the solution.4[2(-10/19) - (3 - (-10/19)) + 5] = -(2 + 7(-10/19))=> 4[-20/19 - (3 + 10/19) + 5] = -(2 - 70/19)=> 4[-20/19 - 57/19 + 5] = -(38/19 - 70/19)=> 4[-77/19 + 5] = -(-32/19)=> 4[-77/19 + 95/19] = 32/19=> 4[18/19] = 32/19=> 72/19 ≠ 32/19Since the left side is not equal to the right side, there is a contradiction.

Key Concepts

Understanding Contradictions in Algebra

In algebra, an equation is a statement that asserts the equality of two expressions. Sometimes, while solving an equation, we come across situations where no possible value of the variable can satisfy the equation. This situation is called a contradiction.
For instance, in the given exercise, the initial equation was simplified to a point where we found that two sides were never equal, no matter what value of x we used.
Specifically, after solving the equation, we got an expression where the left side of the equation was not equal to the right side (i.e., \(72/19 ≠ 32/19\)).
This means our equation has no solution. Such an equation is called a contradiction. Understanding this concept helps us recognize when certain algebraic equations can't be resolved.

Linear Equations

Linear equations are equations that can be written in the form \(Ax + B = C\), where A, B, and C are constants, and x is the variable.
They are called linear because their graph is a straight line. Linear equations are fundamental in algebra and often serve as the building blocks for more complex equations.
In our example, the equation \(4[2x - (3-x) + 5] = -(2 + 7x)\) simplifies to the linear equation \(19x + 8 = -2\), which represents a line if plotted on a graph.
Solving linear equations typically involves isolating the variable on one side of the equation:

• Combine like terms
• Move variables to one side
• Isolate the variable

Finally, we solve for the variable to find its value. However, in our exercise, we discovered that the equation is a contradiction and thus has no solutions.

Identity vs. Contradiction

When solving equations, we may encounter special cases: identities and contradictions. These two concepts are essential to understand:

• An **identity** is an equation that holds true for all values of the variable. For example, the equation \(2(x + 1) = 2x + 2\) is an identity, as simplifying both sides shows they are always equal.

Such equations mean the left-hand side and the right-hand side are identical expressions.

• A **contradiction**, on the other hand, is an equation that has no solution. The simplification shows that no value of the variable can make the equation true.

In the given exercise, after solving all the steps, we found the algebraic equation leading to a statement \(72/19 ≠ 32/19\), demonstrating it’s a contradiction since no possible value of x satisfies it.
Understanding the difference between an identity and a contradiction is crucial in algebra, as it helps in drawing correct conclusions about the nature of the equations we solve.