Final answer:
To create a system of two linear equations with the solution (5, 3), we first develop different equations using arbitrary slopes and the point-slope form. This results in the equations y = 2x - 7 and y = -x + 8, which both satisfy the required solution when checked.
Step-by-step explanation:
To write a system of two linear equations in x and y that has the ordered pair solution (5,3), we can start by creating two different linear equations that both pass through the point (5, 3). A simple way to create such equations is to choose arbitrary values for the slope (m) and the y-intercept (b) for each equation.
For the first equation, let's use the point-slope form of a line which is y - y1 = m(x - x1). If we choose a slope of 2, the equation becomes:
y - 3 = 2(x - 5)
Expanding and simplifying gives us:
y = 2x - 7
For the second equation, we can use the same form but choose a different slope, such as -1, resulting in:
y - 3 = -1(x - 5)
Which simplifies to:
y = -x + 8
Now we have a system of two linear equations:
To verify if (5, 3) is indeed a solution to both equations, we can substitute in the values:
- For y = 2x - 7: 3 = 2(5) - 7, which is true because 3 = 10 - 7 = 3.
- For y = -x + 8: 3 = -(5) + 8, which is also true because 3 = -5 + 8 = 3.