Consider the pair of linear equations below.
4x+6y=12
2x+3y=6
What is the relationship,if any, between the two equations
Does the system of equations have one solution, no solution, or infinitely many solutions.Explain
How can you verify your answers to the two questions above by solving algebraically
Happy Sunday!
Did you notice that the second equation is EXACTLY like the first equation (but the parts are all multiplied by 2).
This means that they are EXACTLY the same equation and lie on top of each other.
If you were to graph both of them, they are the same line : )
Thus an infinite number of solutions.
Thank you very much
4x+6y
To determine the relationship between the two linear equations, let's analyze them together:
4x + 6y = 12 ...(Equation 1)
2x + 3y = 6 ...(Equation 2)
To find the relationship between these equations, we examine their slopes and y-intercepts. We can do this by rewriting the equations in slope-intercept form (y = mx + b), where m represents the slope and b represents the y-intercept.
Equation 1:
4x + 6y = 12
Rearrange to get:
6y = -4x + 12
Divide both sides by 6:
y = (-4/6)x + 2
Simplified, we get:
y = -(2/3)x + 2
Equation 2:
2x + 3y = 6
Rearrange to get:
3y = -2x + 6
Divide both sides by 3:
y = (-2/3)x + 2
Simplified, we get:
y = -(2/3)x + 2
By comparing the two equations, we observe that they have the same slope (-2/3) and the same y-intercept (2). Therefore, the two equations represent the same line in the coordinate plane.
Regarding the number of solutions, when two equations represent the same line, there are infinitely many solutions. Every point on the line satisfies both equations simultaneously.
To verify this algebraically, we can solve the system of equations using any method, such as substitution or elimination:
Let's use the substitution method to solve the system algebraically:
Start with Equation 1:
4x + 6y = 12
Rearrange to solve for x:
4x = 12 - 6y
Divide by 4:
x = (12 - 6y)/4
Simplified, we get:
x = 3 - (3/2)y
Substitute this expression for x into Equation 2:
2(3 - (3/2)y) + 3y = 6
Simplify:
6 - 3y + 3y = 6
The variable y cancels out, resulting in:
6 = 6
This equation is always true, indicating that no matter the value of y, the equation holds. As a result, the system has infinitely many solutions.
Therefore, the relationship between the two equations is that they represent the same line, and the system has infinitely many solutions.