Question

Solve the following equation for $x$ : $19-2x=-1$

Short answer

The short answer is $x=10$.

Step-by-step solution

Rewrite the equation

We are given the equation, $19-2x=-1$.

Add 2x to both sides of the equation

In order to isolate the term containing the variable $x$, we add $2x$ to both sides of the equation: $$19-2x+2x=-1+2x$$ This simplifies to: $$19=-1+2x$$

Add 1 to both sides of the equation

Now, to further isolate the term containing $x$, we add $1$ to both sides of the equation: $$19+1=-1+1+2x$$ This simplifies to: $$20=2x$$

Divide both sides of the equation by 2

Finally, we isolated the term containing $x$, so now we just need to solve for the variable itself. To do this, divide both sides of the equation by $2$: $$\frac{20}{2}=\frac{2x}{2}$$ This simplifies to: $$10=x$$

State the solution

Now, we found the solution for the given equation, which is $x=10$.

Key concepts

Algebraic Equations

Algebraic equations are mathematical statements that indicate the equality of two expressions. They typically involve variables, numbers, and operation symbols. For example, in the equation 19 - 2x = -1, x represents a variable - an unknown quantity we are trying to find.

Understanding the properties of equality is crucial for solving algebraic equations. It allows you to perform the same operation on both sides of the equation without changing its equality. For instance, if you add, subtract, multiply, or divide both sides of an equation by the same number, the equation remains balanced.

It's also essential to recognize different types of equations, such as linear equations, which are the simplest form where variables are to the power of one and graph as straight lines. The represented example is a linear equation. Its solution identifies a point where the line would intersect the x-axis on a graph.

Isolate the Variable

To solve an equation for a particular variable means to isolate that variable on one side of the equation. Isolation is typically done through a series of operations that reverse the operations conducted on the variable. The goal is to get the variable by itself, with a coefficient of one.

In the example equation 19 - 2x = -1, the variable x is initially surrounded by other terms. By performing operations like addition or subtraction, you can systematically move all other numbers to the opposite side of the equation. The use of inverse operations, such as undoing subtraction with addition (or vice versa), and canceling out division with multiplication (or vice versa), are common strategies used during isolation.

Consistency in Isolation

The process of isolating the variable should be consistent, to both maintain the equality of the equation and to logically step towards simplifying it. In the provided example, each step progresses towards having x alone on one side, showing a clear approach to arriving at the solution.

Equation Solving Steps

To effectively solve linear equations, certain systematic steps can be followed. These steps simplify the equation and isolate the variable, a process which was exemplified in the given problem.

The initial steps usually involve simplifying each side of the equation by combining like terms and using the distributive property if necessary. Once simplified, the next set of steps involves moving terms that contain the variable to one side, and constants to the other.

Applying Operations Equally

Every time an operation is applied to one side of the equation, it must also be applied to the other side to maintain equality. This series of operations continues until the variable is isolated. The final steps usually involve division or multiplication to solve for the variable explicitly, after which the solution is stated.

• Rewriting the equation
• Moving variable terms to one side
• Isolating the variable
• Finding the solution

When all operations are correctly applied, the variable is solved for, and the original equation's solution is determined, as shown in the step by step solution for x = 10.