Problem 19 - Elementary Algebra

Question

Solve the equations. $$ 4 x+2=-2 $$

Step-by-Step Solution

Answer

Answer: The value of x is -1.

Step 1: Analyze the equation

The given equation is a linear equation of the form \(ax + b = c\). In this case, \(a = 4\), \(b = 2\), and \(c = -2\). Our task is to find the value of the variable \(x\) that satisfies this equation.

Step 2: Isolate x

To solve for \(x\), we will first get rid of the constant term on the left side of the equation. This can be done by subtracting \(2\) from both sides of the equation: $$ 4x + 2 - 2 = -2 - 2 \\ 4x = -4 $$

Step 3: Solve for x

Now we just need to divide both sides of the equation by \(4\) to find the value of \(x\): $$ \frac{4x}{4} = \frac{-4}{4} \\ x = -1 $$

Step 4: Check the solution

As a final step, we should check that our solution is correct. We will substitute our answer, \(x = -1\), back into the original equation and see if both sides are equal: $$ 4(-1) + 2 = -4 + 2 = -2 $$ The left side of the equation is equal to the right side, so our solution \(x = -1\) is correct.

Key Concepts

Solving Equations

Solving equations is the process of finding the value of unknown variables that make the equation true. In linear equations, like the one we are considering, all variables are raised to the power of one. This means each variable appears only in linear terms, making these equations straightforward to solve. The general form of a linear equation is given by:

• \( ax + b = c \)

where \( a \), \( b \), and \( c \) are constants.
To solve such equations, we carry out operations to the equation that help "simplify" it, ultimately isolating the variable we want to solve for. Each step involves doing identical operations to both sides of the equation to maintain equality. This includes addition, subtraction, multiplication, or division.
The example shows the process of isolating \( x \) through subtraction and division, which are common operations in solving linear equations. Each step gets us closer to knowing the value of \( x \).

Isolation of Variables

Isolation of variables is a crucial technique in solving equations. It involves manipulating the equation in such a way that the variable we are solving for stands alone on one side of the equation. Consider the equation from our example:

• \( 4x + 2 = -2 \)

The first step towards isolation is to remove any constants from the side of the equation where the variable appears. In our case, subtracting 2 from both sides does this:

• \( 4x + 2 - 2 = -2 - 2 \)
• \( 4x = -4 \)

Once the constants are eliminated, any coefficients attached to the variable need to be isolated by applying the appropriate operation. Here, dividing both sides by 4, the coefficient of \( x \), gives:

• \( \frac{4x}{4} = \frac{-4}{4} \)

These operations gradually peel away layers from the equation to reveal the isolated variable, thus solving for \( x \).

Substitution Method

While our example doesn't explicitly use the substitution method, understanding it can be very helpful when dealing with systems of equations. The substitution method involves solving one equation for one variable and then substituting that expression into another equation, allowing us to solve for a second variable. Here's how it typically works:

• Solve one of the equations for one of its variables.
• Substitute this expression for the variable in the other equation.
• Simplify and solve the new equation for the remaining variable.
• Backtrack to find the expression for the first variable.

In single-variable equations, substitution is more about checking solutions, as was done in Step 4 of our process. By plugging the solution back into the original equation, we verify its correctness.
This clarity ensures that the solution satisfies the initial conditions set by the problem. For more complex systems, substitution plays a vital role in streamlining problem-solving.