To find the point where the two equations intersect, we can solve the system of equations:
- \( 4x - 2y = 2 \)
- \( 12x - 3y = 9 \)
Let's first simplify the second equation. We can divide all terms by 3:
\[ 4x - y = 3 \]
Now we have a simpler system of linear equations:
- \( 4x - 2y = 2 \)
- \( 4x - y = 3 \)
Next, let's solve for \( y \) in the second equation:
\[ y = 4x - 3 \]
Now substitute \( y \) from the second equation into the first equation:
\[ 4x - 2(4x - 3) = 2 \]
Distributing the -2:
\[ 4x - 8x + 6 = 2 \]
Combine like terms:
\[ -4x + 6 = 2 \]
Subtract 6 from both sides:
\[ -4x = -4 \]
Dividing both sides by -4 gives:
\[ x = 1 \]
Now that we have \( x \), we can substitute it back into the equation for \( y \):
\[ y = 4(1) - 3 = 4 - 3 = 1 \]
Therefore, the point where the two lines intersect is:
\[ (1, 1) \]
Thus, the complete sentence would read:
When graphed, the two linear equations intersect at the point \( (1, 1) \).