Question

(08.01) Two lines, A and B, are represented by the following equations: Line A: 3x + 3y = 12 Line B: x + y = 4 Which statement is true about the solution to the set of equations? (4 points) Question 2 options: 1) It is (12, 4). 2) There are infinitely many solutions. 3) It is (4, 12). 4) There is no solution.

Answer 1

• Infinitely many solutions.

Explanation:

To find:-

• The correct option from the given ones .

Answer:-

We are here given that there are two linear equations, namely,

`\begin{cases} 3x+3y = 12 \\ x+y = 4 \end{cases}`

These can be rewritten as ,

`\begin{cases} 3x+3y - 12 =0\\ x+y -4=0 \end{cases}`

Before we precede we must know that,

Conditions for solvability :-

If there are two linear equations namely,

`\begin{cases} a_x + b_1y+c_1 = 0 \\ a_2x+b_2y+c_2=0\end{cases}`

Then ,

Case 1 :-

If we have,

`\longrightarrow \boxed{ (a_1)/(a_2)=(b_1)/(b_2)=(c_1)/(c_2) } \\`

Then , the lines are coincident and there are infinitely many solutions .

Case 2 :-

If we have,

`\longrightarrow \boxed{ (a_1)/(a_2)=(b_1)/(b_2)\\eq(c_1)/(c_2) } \\`

Then, the linear equations are inconsistent and have no solutions , thus the lines are parallel .

`\rule{200}2`

So here with respect to angle standard form of pair linear equations, we have;

• 
  `a_1 = 3` ,
  `b_1 = 3` ,
  `c_1 = -12`
• 
  `a_2= 1` ,
  `b_2 = 1` ,
  `c_2 = -4`

Hence here we have,

`\longrightarrow (a_1)/(a_2) = (3)/(1) \\`

`\longrightarrow (b_1)/(b_2)=(3)/(1) \\`

`\longrightarrow (c_1)/(c_2)=(-12)/(-4)=(3)/(1) \\`

Therefore we can clearly see that,

`\longrightarrow \boxed{ (a_1)/(a_2)=(b_1)/(b_2)=(c_1)/(c_2) =\boxed{(3)/(1)}} \\`

Hence there are infinitely many solutions and the lines are coincident .

Answer 2

Explanation:

The solution to the set of equations can be found by solving the system:

3x + 3y = 12

x + y = 4

We can simplify the second equation by solving for y:

y = 4 - x

Substituting this expression for y into the first equation, we get:

The solution to the set of equations can be found by solving the system:

3x + 3y = 12

x + y = 4

We can simplify the second equation by solving for y:

y = 4 - x

Substituting this expression for y into the first equation, we get:

3x + 3(4 - x) = 12

3x + 12 - 3x = 12

12 = 12

This is a true statement, which means that the system is consistent and has infinitely many solutions. Therefore, the correct answer is:

There are infinitely many solutions.