Answer :
Let's solve the system of equations using substitution:
[tex]\[ \begin{cases} 4x + 2y = 6 \\ x = 2y + 4 \end{cases} \][/tex]
Step 1: Substitute [tex]\( x = 2y + 4 \)[/tex] into the first equation.
The first equation is:
[tex]\[ 4x + 2y = 6. \][/tex]
Replace [tex]\( x \)[/tex] with [tex]\( 2y + 4 \)[/tex]:
[tex]\[ 4(2y + 4) + 2y = 6. \][/tex]
Step 2: Simplify and solve for [tex]\( y \)[/tex].
Distribute the [tex]\( 4 \)[/tex]:
[tex]\[ 8y + 16 + 2y = 6. \][/tex]
Combine like terms:
[tex]\[ 10y + 16 = 6. \][/tex]
Isolate [tex]\( y \)[/tex] by subtracting 16 from both sides:
[tex]\[ 10y = 6 - 16, \][/tex]
[tex]\[ 10y = -10. \][/tex]
Divide both sides by 10:
[tex]\[ y = -1. \][/tex]
Step 3: Solve for [tex]\( x \)[/tex].
Substitute [tex]\( y = -1 \)[/tex] back into the second equation:
[tex]\[ x = 2y + 4. \][/tex]
So,
[tex]\[ x = 2(-1) + 4, \][/tex]
[tex]\[ x = -2 + 4, \][/tex]
[tex]\[ x = 2. \][/tex]
Step 4: Write the solution as an ordered pair.
The solution to the system is:
[tex]\[ (x, y) = (2, -1). \][/tex]
Step 5: Verify the solution.
Check the solution [tex]\((2, -1)\)[/tex] in both original equations:
For the first equation:
[tex]\[ 4(2) + 2(-1) = 6, \][/tex]
[tex]\[ 8 - 2 = 6, \][/tex]
[tex]\[ 6 = 6 \quad \text{(True)}. \][/tex]
For the second equation:
[tex]\[ 2 = 2(-1) + 4, \][/tex]
[tex]\[ 2 = -2 + 4, \][/tex]
[tex]\[ 2 = 2 \quad \text{(True)}. \][/tex]
Thus, the solution [tex]\((2, -1)\)[/tex] satisfies both equations. The correct answer from the provided options is:
[tex]\[ (2, -1). \][/tex]
[tex]\[ \begin{cases} 4x + 2y = 6 \\ x = 2y + 4 \end{cases} \][/tex]
Step 1: Substitute [tex]\( x = 2y + 4 \)[/tex] into the first equation.
The first equation is:
[tex]\[ 4x + 2y = 6. \][/tex]
Replace [tex]\( x \)[/tex] with [tex]\( 2y + 4 \)[/tex]:
[tex]\[ 4(2y + 4) + 2y = 6. \][/tex]
Step 2: Simplify and solve for [tex]\( y \)[/tex].
Distribute the [tex]\( 4 \)[/tex]:
[tex]\[ 8y + 16 + 2y = 6. \][/tex]
Combine like terms:
[tex]\[ 10y + 16 = 6. \][/tex]
Isolate [tex]\( y \)[/tex] by subtracting 16 from both sides:
[tex]\[ 10y = 6 - 16, \][/tex]
[tex]\[ 10y = -10. \][/tex]
Divide both sides by 10:
[tex]\[ y = -1. \][/tex]
Step 3: Solve for [tex]\( x \)[/tex].
Substitute [tex]\( y = -1 \)[/tex] back into the second equation:
[tex]\[ x = 2y + 4. \][/tex]
So,
[tex]\[ x = 2(-1) + 4, \][/tex]
[tex]\[ x = -2 + 4, \][/tex]
[tex]\[ x = 2. \][/tex]
Step 4: Write the solution as an ordered pair.
The solution to the system is:
[tex]\[ (x, y) = (2, -1). \][/tex]
Step 5: Verify the solution.
Check the solution [tex]\((2, -1)\)[/tex] in both original equations:
For the first equation:
[tex]\[ 4(2) + 2(-1) = 6, \][/tex]
[tex]\[ 8 - 2 = 6, \][/tex]
[tex]\[ 6 = 6 \quad \text{(True)}. \][/tex]
For the second equation:
[tex]\[ 2 = 2(-1) + 4, \][/tex]
[tex]\[ 2 = -2 + 4, \][/tex]
[tex]\[ 2 = 2 \quad \text{(True)}. \][/tex]
Thus, the solution [tex]\((2, -1)\)[/tex] satisfies both equations. The correct answer from the provided options is:
[tex]\[ (2, -1). \][/tex]
