To determine how many solutions the system of equations has, we can consider the two given equations:
- \( y = 6x + \frac{4}{5} \)
- \( y = \frac{1}{8}x + \frac{5}{9} \)
We set these two equations equal to each other to find any intersection points:
\[ 6x + \frac{4}{5} = \frac{1}{8}x + \frac{5}{9} \]
Next, we'll isolate \(x\) by first eliminating the fractions. To do this, we can find a common denominator for the constants, which can be \(40\) (the least common multiple of denominators \(5\) and \(9\)).
Multiplying the entire equation by \(40\):
\[ 40(6x) + 40\left(\frac{4}{5}\right) = 40\left(\frac{1}{8}x\right) + 40\left(\frac{5}{9}\right) \]
This simplifies to:
\[ 240x + 32 = 5x + \frac{200}{9} \]
Next, we will multiply through by \(9\) to eliminate the fraction:
\[ 9(240x + 32) = 9(5x + \frac{200}{9}) \]
This simplifies to:
\[ 2160x + 288 = 45x + 200 \]
Now, we'll organize the equation by moving \(45x\) to the left:
\[ 2160x - 45x + 288 = 200 \] \[ 2115x + 288 = 200 \]
Next, we isolate \(x\) by subtracting \(288\) from both sides:
\[ 2115x = 200 - 288 \] \[ 2115x = -88 \]
Now, solve for \(x\):
\[ x = \frac{-88}{2115} \]
Finally, we will substitute this value of \(x\) back into one of the original equations to find \(y\). Let's use the first equation:
\[ y = 6\left(\frac{-88}{2115}\right) + \frac{4}{5} \]
To simplify this calculation, we can analyze the equations more succinctly.
Since the coefficients of \(x\) in both equations are different, where one is a slope of \(6\) and the other is \(\frac{1}{8}\), it indicates that the lines intersect at exactly one point.
Thus, the system of equations has:
One solution.
1 answer