What is the equivalent system for the equations 23x + y = 2 and x + 12y = 3?

To determine the equivalent system of the given equations, we first need to analyze the equations:

1. Equation 1: 23x + y = 2
2. Equation 2: x + 12y = 3

Next, we will solve these equations to find an equivalent system that may present the same solution set.

Step 1: Solve Equation 1 for y:
We can rearrange Equation 1 to isolate y:

y = 2 - 23x

Step 2: Substitute y in Equation 2:
Let’s substitute the expression we found for y into Equation 2:

x + 12(2 - 23x) = 3

Now, distribute 12 across the parentheses:

x + 24 - 276x = 3

This simplifies to:

-275x + 24 = 3

Now, isolate -275x:

-275x = 3 - 24

-275x = -21

Dividing both sides by -275 gives:

x = rac{21}{275}

Step 3: Solve for y:
Now substitute x back into the expression for y:

y = 2 - 23(rac{21}{275})

This can be calculated as:

y = 2 - rac{483}{275}

Finding a common denominator:

y = rac{550}{275} - rac{483}{275} = rac{67}{275}

So we have our solution set:

x = rac{21}{275}, y = rac{67}{275}

To construct an equivalent system, we can express these relationships as:

– Multiply Equation 1 by 1 (to keep it the same):
23x + y - 2 = 0

– Multiply Equation 2 by 1 (to keep it the same):
x + 12y - 3 = 0

Thus, the equivalent system is:

This shows how we derived the equivalent system from the given equations.