Question

solve the system of equations below by graphing.\

$$\begin{cases}0.2x - 0.5y = 10\\\\0.5x + 0.3y = 15\\end{cases}$$

\
what is the solution rounded to the nearest hundredth?\
\\( (23.19, -11.34) \\)\
\\( (23.19, 11.34) \\)\
\\( (33.87, -6.45) \\)\
\\( (33.87, 6.45) \\)

Explanation:

Step1: Rewrite equations in slope - intercept form

For the first equation \(0.2x - 0.5y=10\):
Subtract \(0.2x\) from both sides: \(- 0.5y=-0.2x + 10\)
Multiply both sides by \(-2\): \(y = 0.4x-20\)

For the second equation \(0.5x + 0.3y=15\):
Subtract \(0.5x\) from both sides: \(0.3y=-0.5x + 15\)
Divide both sides by \(0.3\): \(y=\frac{-0.5}{0.3}x+\frac{15}{0.3}=-\frac{5}{3}x + 50\)

Step2: Analyze the intersection

We can also solve the system using the elimination method. Multiply the first equation by \(0.3\) and the second equation by \(0.5\) to eliminate \(y\):

First equation after multiplication: \(0.06x-0.15y = 3\)

Second equation after multiplication: \(0.25x+0.15y=7.5\)

Add the two new equations: \((0.06x + 0.25x)+(-0.15y + 0.15y)=3 + 7.5\)

Simplify: \(0.31x=10.5\)

Solve for \(x\): \(x=\frac{10.5}{0.31}\approx33.87\)

Substitute \(x = 33.87\) into the first original equation \(0.2x-0.5y = 10\):

\(0.2\times33.87-0.5y=10\)

\(6.774-0.5y = 10\)

Subtract \(6.774\) from both sides: \(-0.5y=10 - 6.774=3.226\)

Multiply both sides by \(-2\): \(y=- 6.452\approx - 6.45\)

Answer:

\((33.87, - 6.45)\)