To solve the equation [tex]\(4 e^{2+2 x}=x-3\)[/tex] by graphing, follow these steps:
1. Write a system of equations:
[tex]\[
\begin{cases}
y = 4 e^{2+2 x} \\
y = x - 3
\end{cases}
\][/tex]
2. Graph the system. Use a graphing calculator or graphing software to plot each equation:
- The first curve is [tex]\(y = 4 e^{2+2 x}\)[/tex]. This is an exponential function with a base of [tex]\(e\)[/tex], scaled by 4.
- The second line is [tex]\(y = x - 3\)[/tex]. This is a straight line with a slope of 1 and a y-intercept of -3.
3. Identify the solutions. The [tex]\(x\)[/tex]-coordinates of the points where the graphs of the equations intersect are the solutions to the original equation.
After graphing, you will find that the curves intersect at:
The equation [tex]\(4 e^{2+2 x}=x-3\)[/tex] has one solution: [tex]\(x \approx -2.03493\)[/tex].
Thus, the value of [tex]\(x\)[/tex] where the curves intersect, and hence the solution to the equation [tex]\(4 e^{2+2 x}=x-3\)[/tex], is approximately [tex]\(-2.03493\)[/tex].
