Question

The table below shows four systems of equations: System 1 System 2 System 3 System 4 4x − 5y = 2 3x − y = 8 4x − 5y = 2 10x − 7y = 18 4x − 5y = 2 3x − 8y = 4 4x − 5y = 2 10x + 3y = 15 Which pair of systems will have the same solution?

Answer 1

The system b is the same

Answer 2

Answer:

System 1 and System 2 are equivalent.

Step-by-step explanation:

The first and second system have the same solutions, the are equivalent systems of equations. Let's calculate solutions to demonstrate it:

System 1.

$\left \{ {{4x - 5y = 2} \atop { 3x - y = 8}} \right.$

If we multiply the second equations by -5, we can eliminate one variable and find the first solution:

$\left \{ {{4x - 5y = 2} \atop { -15x +5y = -40}} \right.\\-11x=-38\\x=\frac{38}{11}$

Now, we use this value to find the other solution:

$3x - y = 8\\3(\frac{38}{11})-y=8\\\frac{114}{11}-8=y\\ y=\frac{114-88}{11}=\frac{26}{11}$

The solution of the first system is $(\frac{38}{11} ;\frac{26}{11} )$

System 2.

$\left \{ {{4x - 5y = 2} \atop { 10x - 7y = 18}} \right.$

We do the same process than we did before, but this time we have to multiply by $-\frac{5}{7}$:

$\left \{ {{4x - 5y = 2} \atop { -\frac{50}{7}x + 5y = \frac{90}{7} }} \right.\\\frac{28x-50x}{7}=\frac{-90+14}{7}\\-22x=-76\\x=\frac{38}{11}$

Then,

$4x - 5y = 2\\4(\frac{38}{11})-5y=2\\\frac{152}{11}-2=5y\\ 5y=\frac{152-22}{11}\\y=\frac{130}{5(11)}=\frac{26}{11}$

The solution of the second system is  $(\frac{38}{11} ;\frac{26}{11} )$

Therefore, system 1 and system 2 are equivalent.