Problem 73 - Algebra for College Students

Question

Use an algebraic approach to solve each problem. Solve each of the following equations. (a) \(5 x+7=5 x-4\) (b) \(4(x-1)=4 x-4\) (c) \(3(x-4)=2(x-6)\) (d) \(7 x-2=-7 x+4\) (e) \(2(x-1)+3(x+2)=5(x-7)\) (f) \(-4(x-7)=-2(2 x+1)\)

Step-by-Step Solution

Answer

(a) No solutions, (b) Infinitely many solutions, (c) x=0, (d) x=3/7, (e) No solutions, (f) No solutions.

Step 1: Simplify Equation (a)

For the equation \(5x + 7 = 5x - 4\), subtract \(5x\) from both sides. This simplifies to \(7 = -4\), which is not a valid statement. Hence, there are no solutions for \(x\).

Step 2: Simplify Equation (b)

The equation is \(4(x-1) = 4x - 4\). Distribute on the left side to get \(4x - 4 = 4x - 4\). Subtract \(4x\) from both sides, obtaining \(-4 = -4\), a true statement, so any real number \(x\) is a solution.

Step 3: Solve Equation (c)

Equation is \(3(x-4)=2(x-6)\). Distribute to get \(3x - 12 = 2x - 12\). Subtract \(2x\) from both sides: \(x - 12 = -12\). Add 12 to both sides: \(x = 0\).

Step 4: Solve Equation (d)

Equation: \(7x - 2 = -7x + 4\). Add \(7x\) to both sides: \(14x - 2 = 4\). Add 2 to both sides: \(14x = 6\). Divide by 14: \(x = \frac{3}{7}\).

Step 5: Solve Equation (e)

Equation: \(2(x-1) + 3(x+2) = 5(x-7)\). Distribute to get \(2x - 2 + 3x + 6 = 5x - 35\). Simplify to \(5x + 4 = 5x - 35\). Subtract \(5x\): \(4 = -35\), no solutions.

Step 6: Solve Equation (f)

Given equation is \(-4(x-7) = -2(2x+1)\). Distribute to \(-4x + 28 = -4x - 2\). Add \(4x\) to both sides: \(28 = -2\), a false statement, indicating no solutions.

Key Concepts

Solving Equations

When you solve equations in algebra, you're basically trying to find the values of the variables that make the equation true. An equation often consists of two expressions connected by an equals sign. These expressions may contain numbers, variables, and operations like addition or multiplication.
To solve an equation, your goal is to manipulate the equation using algebraic methods to isolate the variable on one side of the equation. Here's how you can do that:

• **Addition or Subtraction:** if something is added to or subtracted from the variable, do the opposite operation to both sides of the equation.


• **Multiplication or Division:** similarly, if the variable is multiplied or divided by a number, perform the inverse operation to both sides.


• **Simplifying Again:** sometimes you'll need to simplify the equation by distributing terms or combining like terms before others steps.

This approach ensures that whatever you do to one side, you must do to the other, preserving the balance of the equation.

Equation Simplification

Equation simplification involves making an equation easier to solve by performing operations that reduce complexity. It's like tidying up your room—everything just makes more sense and is easier to navigate.
To simplify an equation, you usually need to:

• **Distribute Terms:** applies especially when there are parentheses. For example, in the equation \(5(x + 3)\), distribute the 5 to get \(5x + 15\).


• **Combine Like Terms:** terms that have the same variable can be combined. For instance, \(3x + 2x\) simplifies to \(5x\).


• **Reduce Constants:** subtract or add constants to consolidate terms on both sides of the equation.

Through simplification, an equation becomes more straightforward and manageable, paving the way for accurate solutions.

No Solution Equations

No solution equations occur when, no matter what value you substitute for the variable, the equation isn't satisfied. This happens when the simplification of an equation leads to a contradiction or a false statement.
A classic example is when, after simplification, you arrive at an equality that is impossible, like \(7 = -4\). Here's why an equation might have no solution:

• **Contradictory Operations:** Sometimes distributing or combining terms can lead to statements where the two sides aren't equal, indicating the absence of a solution.


• **Identical Terms Canceling:** Equations like \(x + 3 = x - 2\) simplify to \(3 = -2\) when \(x\) is subtracted from both sides, showing no possible solution for \(x\).

Recognizing and understanding no solution equations is essential as it's a part of grasping the broader landscape of algebraic problem-solving.