which is the approximate solution to the system y = 0.5x + 3.5 and y = -\\frac{2}{3}x+\\frac{1}{3} shown on…

Explanation:

Step1: Set the two equations equal

Since at the solution point $y$ - values are equal, we set $0.5x + 3.5=-\frac{2}{3}x+\frac{1}{3}$.

Step2: Convert decimals to fractions

$0.5=\frac{1}{2}$, so the equation becomes $\frac{1}{2}x + 3.5=-\frac{2}{3}x+\frac{1}{3}$. And $3.5=\frac{7}{2}$, then $\frac{1}{2}x+\frac{7}{2}=-\frac{2}{3}x+\frac{1}{3}$.

Step3: Get a common - denominator

The common denominator of 2 and 3 is 6. Multiply each term by 6: $6\times\frac{1}{2}x+6\times\frac{7}{2}=6\times(-\frac{2}{3}x)+6\times\frac{1}{3}$. This simplifies to $3x + 21=-4x + 2$.

Step4: Solve for $x$

Add $4x$ to both sides: $3x+4x + 21=-4x+4x + 2$, which gives $7x+21 = 2$. Then subtract 21 from both sides: $7x+21 - 21=2 - 21$, so $7x=-19$, and $x=-\frac{19}{7}\approx - 2.7$.

Step5: Solve for $y$

Substitute $x\approx - 2.7$ into $y = 0.5x+3.5$. $y=0.5\times(-2.7)+3.5=-1.35 + 3.5 = 2.15\approx2.1$.

Answer:

$(-2.7,2.1)$