Question

How many solutions does the system of equations below have? y=6x+4/5 y=1/8x+5/9

Answer 1

To determine how many solutions the system of equations has, we can consider the two given equations:

1. \( y = 6x + \frac{4}{5} \)
2. \( y = \frac{1}{8}x + \frac{5}{9} \)

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.**