Question

Solve the equation. $$-10=\frac{1}{2} x+x$$

Short answer

The solution to the equation is \( x = -6.67 \) or \( x = -\frac{20}{3} \).

Step-by-step solution

Simplify the Equation

First, look at the expression on the right-hand side of the equation \( -10 = \frac{1}{2} x + x \). Here, it's important to note that \( x \) can be considered as \( 2/2 x \). So, the equation can be rewritten as \( -10 = \frac{1}{2} x + \frac{2}{2} x = \frac{3}{2} x \). So the simplified equation is \( -10 = \frac{3}{2} x \).

Isolate the Variable

To find the value of the variable \( x \), we need to get rid of the coefficient of \( x \) on the right-hand side. This can be achieved by multiplying both sides of the equation by the reciprocal of \( 3/2 \), which is \( 2/3 \). So the equation becomes \( -10 * \frac{2}{3} = x \).

Evaluate the Value of x

Now just solve \( -10 * \frac{2}{3} \) to find the value of \( x \). Simplifying, we get \( x = -\frac{20}{3} \) or \( x = -6.67 \).

Key concepts

Solving Equations Step by Step

Mastering the process of solving equations step by step is a cornerstone of algebra. Grasping this process empowers students to tackle a wide variety of mathematical challenges. Let's consider our initial equation:
\[-10=\frac{1}{2} x+x\]
The key to solving this step by step is to address each part of the equation systematically. Start with identifying like terms and combining them. Then, streamline the equation so it's simpler to manage. A critical point here is maintaining equilibrium - whatever operation you perform on one side, do the same to the other to keep the equation balanced. Understanding that the ultimate goal is to isolate the variable will guide your operations throughout these steps.
Additionally, checking your work is invaluable. By substituting your solution back into the original equation, you verify the correctness of your steps. This not only boosts confidence in your solution but also deepens your comprehension of the relationships between the terms in an equation.

Equation Simplification

Simplifying an equation makes it easier to see the path to the solution. Simplification often involves combining like terms, reducing fractions, or factoring. Consider the provided exercise where we combine the terms with the variable \(x\):
\[\frac{1}{2} x + x = \frac{1}{2} x + \frac{2}{2} x\].
In this step, we're essentially converting all terms to have the same denominator to combine them effectively, which is a common technique when dealing with fractions. Simplification can sometimes also involve expanding expressions, or utilizing properties of operations, like the distributive property. The clearer and simpler the equation, the less room for error when you move to isolate the variable and solve.

Isolate the Variable

Isolating the variable is the most crucial step in solving a linear equation. It involves manipulating the equation to get the variable on one side and everything else on the other. This means undoing anything that’s being done to the variable. In our example, \(\frac{3}{2}x = -10\), the variable \(x\) is being multiplied by \(\frac{3}{2}\).
To undo this multiplication, we use the inverse operation—division. Since we cannot directly divide by a fraction, we multiply by its reciprocal. Here, multiplying both sides by \(\frac{2}{3}\) isolates \(x\):
\[-10 \times \frac{2}{3} = x\].
It's essential to remain vigilant about keeping the equation balanced by treating both sides equally. Isolating the variable lays everything out clearly, so you can confidently derive the solution. When the variable stands alone, you have found the solution to the equation.