Problem 61 - Elementary Algebra

Question

Solve. $$ -x+5=14 $$

Step-by-Step Solution

Answer

x = -9

Step 1: Isolate the variable term

First, we need to isolate the term with the variable on one side of the equation. Currently, we have \(-x + 5 = 14\).Subtract 5 from both sides to remove the 5 away from the left side:\(-x + 5 - 5 = 14 - 5\). This simplifies to:\(-x = 9\).

Step 2: Solve for x

Since we have \(-x = 9\), we need to solve for positive \(x\). We do this by multiplying both sides by \(-1\) to change the sign of \(x\):\(-1 \times (-x) = -1 \times 9\).This results in:\(x = -9\).

Key Concepts

Isolate the Variable

In solving linear equations, one of the most important steps is isolating the variable. The goal here is to have the variable, like \(x\), all by itself on one side of the equation. This makes it easier to solve for its value.

• To "isolate the variable," you need to remove any constants or coefficients that are attached to it by using opposite operations.
• For example, in the equation \(-x + 5 = 14\), \(x\) is combined with 5. You need to move the 5 to the other side of the equation by subtracting 5 from both sides.
• This gives you a simpler equation: \(-x = 9\).

By isolating the variable, you transform a more complicated equation into a simple one where you can clearly see what must be done next to solve for the variable. This is a crucial step in problem-solving not just in math class but in many real-world applications too.

Equation Balancing

Equation balancing is like a mathematical seesaw. To keep the equation balanced, any operation you perform on one side must also be performed on the other side. This principle ensures that the two sides of the equation remain equal throughout your calculation.

• If you add, subtract, multiply, or divide a number on one side, do exactly the same to the other side.
• In our example, \(-x + 5 = 14\), we subtracted 5 from both sides, maintaining the balance of the equation: \(-x + 5 - 5 = 14 - 5\), simplifying to \(-x = 9\).
• This consistency is key to correctly solving equations without altering their solutions or meaning.

Practicing this balancing act helps strengthen our understanding of equality in equations, ensuring we can apply this skill to more complex math problems in the future.

Multiplying by Negative One

In the process of solving equations, sometimes you'll find a negative sign attached to the variable. When this happens, you may need to multiply both sides of the equation by negative one to change the sign of the variable.

• This is important because you usually want the variable to be positive before stating the solution.
• In our example, we ended up with \(-x = 9\). Multiplying both sides by \(-1\) is necessary to "flip" the sign of \(x\).
• Doing this calculation, \(-1 \times (-x) = -1 \times 9\), gives us \(x = -9\).

Changing the sign of the variable through multiplication helps turn a problem with subtraction or negative factors into one that is easier to understand and solve. This not only makes the equations simpler but also aligns with most standard practices in algebra for presenting final answers.