Sure, let's solve the system of equations step by step to find the correct representation and the solution. The system of equations given is:
1. [tex]\( x - 2y = 5 \)[/tex]
2. [tex]\( 3x + 15y = -6 \)[/tex]
First, let’s solve these equations simultaneously.
### Step 1: Convert them into a standard form
The equations are already in a usable form:
- Equation 1: [tex]\( x - 2y = 5 \)[/tex]
- Equation 2: [tex]\( 3x + 15y = -6 \)[/tex]
### Step 2: Solve Equation 1 for [tex]\( x \)[/tex]
From the first equation:
[tex]\[ x = 2y + 5 \][/tex]
### Step 3: Substitute this expression into Equation 2
Substituting [tex]\( x \)[/tex] from Equation 1 into Equation 2:
[tex]\[ 3(2y + 5) + 15y = -6 \][/tex]
### Step 4: Simplify and solve for [tex]\( y \)[/tex]
Distributing the 3 in the second equation:
[tex]\[ 6y + 15 + 15y = -6 \][/tex]
Combine like terms:
[tex]\[ 21y + 15 = -6 \][/tex]
Subtract 15 from both sides:
[tex]\[ 21y = -21 \][/tex]
Divide by 21:
[tex]\[ y = -1 \][/tex]
### Step 5: Solve for [tex]\( x \)[/tex]
Now substitute [tex]\( y = -1 \)[/tex] back into the expression for [tex]\( x \)[/tex]:
[tex]\[ x = 2(-1) + 5 \][/tex]
[tex]\[ x = -2 + 5 \][/tex]
[tex]\[ x = 3 \][/tex]
So, the solution to the system of equations is:
[tex]\[ (x, y) = (3, -1) \][/tex]
### Conclusion
Therefore, the ordered pair [tex]\( (3, -1) \)[/tex] is the solution to the system, and the graph that correctly represents the given system of equations should intersect at this point [tex]\( (3, -1) \)[/tex].
To complete the sentence:
"The graph that correctly represents the given system of equations is graph [tex]\( \square \)[/tex] and the solution to the system is ( [tex]\(3, -1\)[/tex] )"
You will need to refer to the graphs presented (Graph A, B, C, D) to identify which graph intersects at [tex]\( (3, -1) \)[/tex].
