Question

mathematics—session 2: 29. a system of equations is given. \\(\

$$\begin{cases} 5x - 8y = 40 \\\\ -1.25x + 2y = -10 \\end{cases}$$

\\) - use the elimination method to determine how many solutions the system has, if any. - show your work and justify your response. - be sure to include your solutions, if any, in your explanation. enter your solutions, your work, and your justifications in the box provided.

Explanation:

Step1: Scale the second equation

Multiply the second equation by 4 to align coefficients with the first equation:
$$4\times(-1.25x + 2y) = 4\times(-10)$$
$$-5x + 8y = -40$$

Step2: Add to first equation

Add the scaled equation to the first equation to eliminate variables:
$$(5x - 8y) + (-5x + 8y) = 40 + (-40)$$
$$0 = 0$$

Step3: Interpret the result

The equation $0=0$ is always true, meaning the two original equations are equivalent (they represent the same line). All points on the line satisfy both equations.

Answer:

This system has infinitely many solutions. Any point $(x, y)$ that satisfies $5x - 8y = 40$ (or equivalently $-1.25x + 2y = -10$) is a solution. We can express the solutions parametrically as $x = t$, $y = \frac{5t - 40}{8}$ where $t$ is any real number.