To find an equation that has infinitely many solutions with the equation \( 5x - 2y = 8 \), we can rewrite this equation in slope-intercept form (which is \( y = mx + b \)).
Starting with the original equation:
\[ 5x - 2y = 8 \]
We can rearrange it to solve for \( y \):
- Subtract \( 5x \) from both sides:
\[ -2y = -5x + 8 \]
- Divide each term by \(-2\):
\[ y = \frac{5}{2}x - 4 \]
The slope-intercept form of the equation is \( y = \frac{5}{2}x - 4 \).
To have a system of equations with infinitely many solutions, we need to create another equation that is equivalent to this one. This means that the new equation can be obtained by multiplying the entire original equation by any non-zero constant.
Let's multiply the original equation \( 5x - 2y = 8 \) by 2:
\[ 2(5x - 2y) = 2(8) \]
This simplifies to:
\[ 10x - 4y = 16 \]
Now let's put it in slope-intercept form:
- Rearranging gives us:
\[ -4y = -10x + 16 \]
- Dividing by \(-4\):
\[ y = \frac{10}{4}x - 4 \]
This simplifies to:
\[ y = \frac{5}{2}x - 4 \]
This equation is identical to our original transformation. Hence, the equation \(10x - 4y = 16\) will have infinitely many solutions with \(5x - 2y = 8\).
So, you can write the infinite solution equation as:
\[ 10x - 4y = 16 \]
Or, equivalently,
\[ y = \frac{5}{2}x - 4 \]