Problem 98 - Elementary Algebra Within Reach

Question

Determine whether (a) \(x=-1\) or (b) \(x=2\) is a solution of the equation. $$-2(x-1)=2-2 x$$

Step-by-Step Solution

Answer

Both \(x=-1\) and \(x=2\) are solutions of the equation \(-2(x-1)=2-2x\).

Step 1: Substitute \(x=-1\) in the equation

The left side of the equation becomes, when \(x=-1\), \(-2(-1-1)\) which simplifies to \(-2(-2) = 4\). The right side of the equation becomes, when \(x=-1\), \(2 - 2(-1)\) which simplifies to \(2+2 = 4\). As both sides of the equation equals 4, \(x=-1\) is a solution of the equation.

Step 2: Substitute \(x=2\) in the equation

The left side of the equation when \(x=2\) is \(-2(2-1)= -2\). The right side of the equation when \(x=2\) is \(2 - 2(2) = -2\). As both sides of the equation equals -2, \(x=2\) is also a solution of the equation.

Key Concepts

Equation Solution

Solving an equation means finding the value(s) of the variable that make the equation true. An equation is like a balance scale—whatever you do to one side, you must do to the other. This maintains the equality.
To check if a number is the solution, simply replace the variable with the number in the equation. After substitution, solve both sides to see if they equal each other. If they do, the number is a solution. If not, it isn't.
For example, in the equation \(-2(x-1) = 2 - 2x\), when we replace \(x\) with a number, we see if both sides result in the same value. If they do, this means our chosen number is a solution of the equation.

Substitution Method

The substitution method is all about replacing variables with numbers to see if they satisfy the equation. It's a crucial step in checking whether a given value is a solution.
Steps for Using Substitution Method:

• Replace the variable in the equation with the given number.
• Simplify the equation to find the result on both sides.
• Compare values of both sides to determine equality.

Let's take an example from the exercise: We substitute \(x = -1\) in the equation \(-2(x-1) = 2 - 2x\). By simplifying, we find both sides equal 4. This shows that \(-1\) is indeed a solution.

Equation Verification

After using substitution, verification ensures that the substitution method was done correctly. It's like a double-check. By comparing the results of both sides of the equation, you confirm whether the number substituted actually makes the equation true.
Steps for Verifying:

• Substitute the number into the equation.
• Calculate the value for each side of the equation separately.
• Ensure both sides of the equation have the same result.

For instance, in our original problem, verification was demonstrated twice. When substituting \(x = 2\), both sides equaled \(-2\). This means our calculations were correct and that \(x = 2\) satisfies the equation.