y = 3x + 6
y = -13x - 4
One Solution
No Solution
Infinite Solution
No Solution
To determine whether the given system of equations has one solution, no solution, or infinite solutions, we can solve the system of equations using the method of substitution or the method of elimination.
Let's use the method of elimination:
Given equations:
y = 3x + 6 --------(1)
y = -13x - 4 --------(2)
To solve the system of equations by elimination, we want to eliminate one variable (either x or y) by manipulating the equations in a way that when we add or subtract them, one variable gets canceled out, leaving us with an equation that only has one variable.
In this case, we can start by multiplying equation (1) by -1:
-1(y = 3x + 6) => -y = -3x - 6
Now we have:
-y = -3x - 6 --------(3)
y = -13x - 4 --------(2)
Next, we add equation (2) and equation (3) together:
(y + y) + (-13x - 3x) + (-4 - (-6))
This simplifies to:
2y + (-16x) + (2) = 0
2y - 16x + 2 = 0
Now, we can rearrange the equation to isolate y:
2y = 16x -2
Dividing the entire equation by 2 gives us:
y = 8x - 1
So, we have the combined equation:
y = 8x - 1 --------(4)
Comparing equation (1) and equation (4), we can see that they are not the same.
Therefore, the system of equations does not have a unique solution, which means the answer is:
No solution
To determine whether the system of equations has one solution, no solution, or infinite solutions, we can compare the slopes and y-intercepts of the two equations.
The given equations are in the form of y = mx + b, where m represents the slope and b represents the y-intercept.
Equation 1: y = 3x + 6
Slope of equation 1: 3
Y-intercept of equation 1: 6
Equation 2: y = -13x - 4
Slope of equation 2: -13
Y-intercept of equation 2: -4
If the slopes of the two lines are different, then the system will have one solution. If the slopes are equal and the y-intercepts are different, then the system will have no solution. If the slopes and y-intercepts are equal, then the system will have infinite solutions.
Comparing the slopes, we see that the slope of equation 1 is 3 and the slope of equation 2 is -13. Since the slopes are different, the system of equations has one solution.