A system of equations can be solved algebraically and graphically. Try both methods to identify their advantages. Which one do you prefer?
Solutions of the system: (-3,-3) and (- 32,- 32) Which method do you prefer? See solution.
Practice makes perfect
We are asked to solve the system of linear and quadratic equations shown below by using two different methods.
y=x & (I) y=2x^2+10x +9 & (II)
We will show how to solve it algebraically and graphically. Then, we will discuss each method's advantages to decide which one is preferred.
Solving the System Algebraically
First of all, note that the y-variable is already isolated in Equation (I). For this reason, it is convenient to start by using the Substitution Method.
Now we can solve the quadratic equation with what we just obtained.
2x^2+9x +9 = 0
Note that it is of the form ax^2+bx+c=0.
ax^2+ bx +c=0
2x^2+ 9x +9 = 0
Therefore, we can factor the equation by finding factors of ac = 18 that add up to b= 9. We can use a table to organize the possibilities.
Factors of 18
Sum
18,1
19 *
9,2
11 *
6,3
9 ✓
With this in mind we can rewrite the x-term on the left-hand side of the equation as 3x+6x and factor the expression by grouping. Once the equation is written in its factored form, we can state its solutions using the Zero Product Property.
The last step to find the solutions of the system is to evaluate any of the functions at the x-values found above. For simplicity we can evaluate y=x.
y = x [1em]
if x =-3, then y =-3 [0.5em]
if x =- 3/2, then y =- 3/2
Therefore, the solutions for the system are (-3,-3) and (- 32,- 32).
Solving the System Graphically
To solve the system by graphing, we need to graph both equations together. The intersection points represent the solutions to the system.
From the graph we can see that one solution is (-3,-3). The other solution does not have integer values, but we can approximate it as (- 32,- 32).
Conclusions
Preferences may vary. However, note that the algebraic method does not require graphing the equations, which can be difficult if you do not have a graphing calculator. Furthermore, it allows us to find the exact solutions every time. When solving the system graphically, sometimes we can only get an approximated solution.
