We start with the two equations provided:
- \( x = 5y + 3 \) (Equation 1)
- \( 3x - 15y = 9 \) (Equation 2)
First, we can substitute Equation 1 into Equation 2. We'll replace \( x \) in Equation 2 with \( 5y + 3 \):
\[ 3(5y + 3) - 15y = 9 \]
Now let's simplify this:
\[ 15y + 9 - 15y = 9 \]
The \( 15y \) and \( -15y \) cancel out:
\[ 9 = 9 \]
This statement is always true, which indicates that the two equations are dependent and represent the same line. Therefore, there are infinitely many solutions.
To express a solution as an ordered pair, we can take Equation 1, which gives us \( x \) in terms of \( y \):
\[ x = 5y + 3 \]
We can choose any value for \( y \), and then calculate the corresponding \( x \).
For example, if we let \( y = 0 \):
\[ x = 5(0) + 3 = 3 \]
So one solution as an ordered pair is:
\[ (3, 0) \]
If we let \( y = 1 \):
\[ x = 5(1) + 3 = 8 \]
Another solution as an ordered pair is:
\[ (8, 1) \]
In general, the solutions can be expressed as:
\[ (x, y) = (5y + 3, y) \quad \text{for any real number } y. \]
Thus, the equations represent a line with infinitely many solutions, with specific examples being \( (3, 0) \) and \( (8, 1) \).
3 answers