Question

example 3: systems with infinitely many solutions:
solve by graphing. \\( 2x + 4y = 8 \\)
\\( y = -\frac{1}{2}x + 2 \\)
why are there infinite ordered pair solutions?
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Explanation:

Step1: Rewrite first equation to slope-intercept form

Start with $2x + 4y = 8$. Subtract $2x$ from both sides:
$4y = -2x + 8$
Divide all terms by 4:
$y = -\frac{2}{4}x + \frac{8}{4}$
Simplify:
$y = -\frac{1}{2}x + 2$

Step2: Compare the two equations

The second given equation is $y = -\frac{1}{2}x + 2$, which is identical to the rewritten first equation.

Step3: Interpret graph meaning

When two linear equations in a system are identical, their graphs are the exact same line. Every point on this line is a solution to both equations.

Answer:

The two equations simplify to the same linear equation ($y = -\frac{1}{2}x + 2$), so their graphs are the same line. Every point on this line represents an ordered pair $(x,y)$ that satisfies both equations, resulting in infinitely many solutions.