To solve the given system of equations by graphing, we need to look at the points where the graphs of the two equations intersect. The given system of equations is:
[tex]\[
\begin{cases}
y = x^2 - 3.5x + 2.2 \\
y = 2x - 5.3625
\end{cases}
\][/tex]
1. Graphing the equations:
- The first equation is a quadratic equation [tex]\( y = x^2 - 3.5x + 2.2 \)[/tex]. This graph will be a parabola opening upwards because the coefficient of [tex]\( x^2 \)[/tex] is positive.
- The second equation is a linear equation [tex]\( y = 2x - 5.3625 \)[/tex]. This graph will be a straight line with a slope of 2 and a y-intercept at [tex]\(-5.3625\)[/tex].
2. Finding the intersection points:
By graphing both equations on the same coordinate plane, we'll look for points where the curve of the parabola intersects the straight line. These intersection points are the solutions to the system of equations.
3. Calculating the intersection points:
Since we need to solve for the x-value where both equations are equal, we set the right-hand sides of each equation equal to each other:
[tex]\[ x^2 - 3.5x + 2.2 = 2x - 5.3625 \][/tex]
4. Solving the equation:
Rearrange the equation to set it to zero:
[tex]\[ x^2 - 3.5x - 2x + 2.2 + 5.3625 = 0 \][/tex]
[tex]\[ x^2 - 5.5x + 7.5625 = 0 \][/tex]
Solve for [tex]\( x \)[/tex] using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = -5.5 \)[/tex], and [tex]\( c = 7.5625 \)[/tex].
5. Finding the solutions:
After solving the quadratic equation, we get the [tex]\( x \)[/tex]-values. We need to plug these [tex]\( x \)[/tex]-values back into either of the original equations to find the corresponding [tex]\( y \)[/tex]-values.
6. Checking the solution:
It turns out that the solution to this system of equations, when the graphs of the parabola and the line intersects, results in the point:
[tex]\[
(2.8, 0.1)
\][/tex]
7. Final answer:
Therefore, the correct rounded intersecting solution from the list of given options is:
[tex]\[
(2.8, 0.1)
\][/tex]
