To set up the given statement as a system of equations, we first need to translate the expressions into mathematical equations.
The original expression states:
- one-fifth of x plus two
- is equal to
- one-third of x plus eight
We can express this in mathematical terms:
- One-fifth of x can be written as: \( \frac{1}{5} x \)
- Two is simply the number 2.
- One-third of x can be written as: \( \frac{1}{3} x \)
- Eight is simply the number 8.
Based on this, we can create the following equation:
\( \frac{1}{5} x + 2 = \frac{1}{3} x + 8 \)
To represent this equation in a system format, we can rearrange it into a standard form of linear equations. To eliminate the fractions, we can multiply through by a common denominator, which in this case is 15:
- Multiply both sides by 15:
- \( 15 \left( \frac{1}{5} x + 2 \right) = 15 \left( \frac{1}{3} x + 8 \right) \)
This simplifies to:
- \( 3x + 30 = 5x + 120 \)
Now we can further rearrange it:
- Bringing all terms involving x on one side and constant terms on the other gives:
- \( 3x – 5x = 120 – 30 \)
- which simplifies to:
- \( -2x = 90 \)
- Thus, we can express the equation in standard form as:
\( 2x + 90 = 0
Now, since we want a system of equations, we can introduce another equation that complements our existing equation. We can use any arbitrary equation to form a system. For example:
- Let’s add: \( y = 2x + 1 \)
Therefore, the system of equations is:
- \( 2x + 90 = 0 \)
- \( y = 2x + 1 \)
In summary, we set up the problem:
- Equation 1: \( 2x + 90 = 0 \)
- Equation 2: \( y = 2x + 1 \)
This system can now be solved using methods such as substitution or elimination, depending on your preference. Each equation represents a relationship between the variables and can be visualized graphically as lines in a Cartesian plane.