Problem 45 - Algebra A Combined Function

Question

Solve. See Examples 1 through 7 $$ 0.7 x-2.3=0.5 $$

Step-by-Step Solution

Answer

The solution is \(x = 4\).

Step 1: Isolate the Variable Term

Start by getting the variable term on one side of the equation. To do this, add 2.3 to both sides of the equation:\[ 0.7x - 2.3 + 2.3 = 0.5 + 2.3 \]This simplifies to:\[ 0.7x = 2.8 \]

Step 2: Solve for the Variable

Next, solve for \(x\) by dividing by 0.7 on both sides of the equation:\[ \frac{0.7x}{0.7} = \frac{2.8}{0.7} \]Which simplifies to:\[ x = 4 \]

Key Concepts

Isolation of Variables

Isolation of variables is a crucial step in solving equations, especially when dealing with linear equations. The goal is to get the variable on one side of the equation by itself, making it easier to solve the equation. For example, consider the equation

• \(0.7x - 2.3 = 0.5\)

To isolate \(x\), you need to address the operation attached to \(x\). In this case, it is important to remove the constant

• \(-2.3\) from the side of the \(x\) term by adding \(2.3\) to both sides.

Performing this isolation step

• \(0.7x = 2.8\)

allows you to proceed with further operations to solve for \(x\). Remember, whatever operation you do to one side of the equation, you must do to the other as well to maintain equality.

Linear Equations

Linear equations are mathematical expressions that create a straight line when graphed on a coordinate plane. Such equations usually take the form:

• \(ax + b = c\)

where \(a\), \(b\), and \(c\) are constants and \(x\) is the variable. In solving linear equations, the objective is to find the value of \(x\) that makes the equation true. For instance, in the problem

• \(0.7x - 2.3 = 0.5\),

the coefficients and constants can be manipulated through basic operations. These operations might include addition, subtraction, multiplication, or division. The beauty of linear equations is that their solutions often can be found through systematic steps, such as isolating the variable and performing inverse operations. Remember, linear equations are named due to their linear relation which is devoid of any exponents or roots.

Algebraic Manipulation

Algebraic manipulation is the technique of rearranging and simplifying equations using arithmetic operations and algebraic properties. It plays a critical role in solving equations like linear equations. For example, to solve

• \(0.7x - 2.3 = 0.5\),

we employ algebraic manipulation by adding \(2.3\) to both sides.
Once the variable term is isolated

• \(0.7x = 2.8\),

the next algebraic manipulation is division to solve for the variable \(x\). Division by the coefficient 0.7 across both sides yields

• \(x = 4\).

In algebraic manipulation, understanding how each basic arithmetic operation affects an equation is key. These skills can simplify complicated equations and make finding solutions much easier.