Compare the number of steps needed to solve the system using each method.
Substitution, see solution.
Practice makes perfect
We want to solve the system of linear equation using all three methods we know.
5x+y=8 & (I) 2y=-10x+8 & (II)
Let's start by solving the system by graphing.
Graphing
To solve a system by graphing, we will start by rewriting both equations into slope-intercept form.
Now we will graph the lines. To do so, we will start by plotting their y-intercepts. For Equation 1 it is point (0, 8), and for Equation 2 it is (0, 4).
Now, we can see that both equations have a slope of - 5. This means that we can start in each y- intercept and move 1 unit right and 5 units down to draw another point that will lie on each equation.
Since the lines have the same slope they are parallel. Therefore, they do not intersect and the system described by these lines has no solution.
We obtain a statement that is always false. This means that the system has no solution.
Elimination
Next let's use elimination. First, let's move all of the variable terms to the left-hand side of Equation 2. To do this we will add 10x to both sides of Equation 2.
5x+y=8 2y=-10x+8 ⇒ 5x+y=8 10x+2y=8
We can multiply Equation 1 by -2. By doing this, variable terms will have opposite coefficients.
We end with a statement that is never true. This means that the system has no solution.
Conclusion
Let's compare the number of steps needed to obtain the solution for each method.
Graphing
Substitution
Elimination
1
Rewriting equations
Solving Equation 2 for y
Rewriting Equation 2
2
Plotting y-intercepts
Substituting the value of y into Equation 1
Multiplying Equation 1 by -2
3
Using slopes to find another point
Solving Equation 1
Adding the equations
4
Connecting points with lines
Solving Equation 1
We can see that solving by substitution requires less steps than the other two methods. This means that we could prefer this method. However, we should always choose a method with which we feel comfortable.
