Solve the following system of equations by graphing. If the system is inconsistent or the equations are dependent, say so
5x-2y=3
10x-4y=6
I know the equations are dependent but I am confused on what is the solution set I'm thinking its (10x-4y=6) but I'm really not sure.
if they are dependent they are the same line. So, the solution set is all real numbers. Any solution to one equation is a solution to the other.
Notice that when you double the first equation (line), you get the second equation.
Thus you have really one equation.
So there would be an infinite number of solutions, that is, any point on the line is a solution
If the second equation would have been
10x - 4y = 5, you would have two separate but parallel lines, which would never intersect.
In that case there would have been no solution.
To solve this system of equations by graphing, we need to plot the lines represented by the equations and find the point where the lines intersect, if they do intersect.
First, let's rearrange the equations in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept:
Equation 1: 5x - 2y = 3
-2y = -5x + 3
y = (5/2)x - 3/2
Equation 2: 10x - 4y = 6
-4y = -10x + 6
y = (10/4)x - 6/4
y = (5/2)x - 3/2
As you can see, both equations have the exact same equation for y. This means that the lines are identical and will overlap each other. Thus, the system of equations is dependent.
The solution set for this dependent system is infinite since every point on the line is a solution to both equations.
Therefore, the solution set for this system of equations is all points on the line given by the equation 10x - 4y = 6, or in slope-intercept form y = (5/2)x - 3/2.
To solve the system of equations by graphing, we need to plot the lines represented by each equation on a graph and find the point of intersection, if it exists.
Let's start by rearranging each equation in the slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept.
Equation 1: 5x - 2y = 3
Rearranging, we get: -2y = -5x + 3
Divide both sides by -2: y = (5/2)x - (3/2)
Equation 2: 10x - 4y = 6
Rearranging, we get: -4y = -10x + 6
Divide both sides by -4: y = (10/4)x - (6/4)
Simplifying, we have: y = (5/2)x - (3/2)
By comparing the two equations, we can observe that they have the same slope (5/2) and the same y-intercept (-3/2). This indicates that the lines are identical and overlap each other, which means they are dependent equations. Therefore, the system of equations has infinitely many solutions.
The solution set can be expressed as a single equation, which in this case is either 5x - 2y = 3 or 10x - 4y = 6. Both equations represent the same line and yield equivalent solutions.