Find the intersection points of the line y = -4x + 2.5 and the curve y = 3.34x/(1 + 2.34x).

Step-by-step solution

1. 1

   Set the two expressions for $y$ equal. At an intersection both formulas give the same height, so

   $-4x+2.5=\frac{3.34x}{1+2.34x}.$

   Record the restriction $x\ne-\frac{1}{2.34}\approx-0.4274$, where the rational curve has a vertical asymptote — any root landing there would have to be discarded.

2. 2

   Clear the denominator. Multiplying both sides by $1+2.34x$:

   $(-4x+2.5)(1+2.34x)=3.34x.$

   Expanding the left side term by term:

   $-4x-9.36x^{2}+2.5+5.85x=-9.36x^{2}+1.85x+2.5.$

   The cross terms are $-4x\cdot2.34x=-9.36x^{2}$ and $2.5\cdot2.34x=5.85x$; combining $-4x+5.85x$ gives the $1.85x$.

3. 3

   Collect into a standard quadratic. Moving $3.34x$ across:

   $-9.36x^{2}+1.85x-3.34x+2.5=0\;\Longrightarrow\;-9.36x^{2}-1.49x+2.5=0,$

   and multiplying by $-1$ to make the leading coefficient positive:

   $9.36x^{2}+1.49x-2.5=0.$

4. 4

   Apply the quadratic formula. With $a=9.36$, $b=1.49$, $c=-2.5$:

   $\Delta=1.49^{2}-4(9.36)(-2.5)=2.2201+93.6=95.8201,\qquad\sqrt{\Delta}=9.788774,$

   $x=\frac{-1.49\pm9.788774}{18.72}\ \Longrightarrow\ x\approx0.443311\ \text{or}\ x\approx-0.602499.$

   Neither equals the excluded value $-0.4274$, so both survive.

5. 5

   Find the $y$-values and check them in the rational equation. Using the line, $y=-4x+2.5$:

   $x=0.443311\Rightarrow y=0.726758,\qquad x=-0.602499\Rightarrow y=4.909994.$

   Substituting into the curve instead: $\frac{3.34(0.443311)}{1+2.34(0.443311)}=0.726758$ ✓ and $\frac{3.34(-0.602499)}{1+2.34(-0.602499)}=4.909994$ ✓. Checking against the other equation, not the one used to compute $y$, is what makes this a real verification.