To solve the system of linear equations
[tex]\[2x + y = 6\][/tex]
[tex]\[4x + 3y = 14\][/tex]
we proceed with the following steps:
1. Express [tex]\(y\)[/tex] from the first equation:
[tex]\[2x + y = 6 \][/tex]
Subtract [tex]\(2x\)[/tex] from both sides to solve for [tex]\(y\)[/tex]:
[tex]\[ y = 6 - 2x \][/tex]
2. Substitute [tex]\(y\)[/tex] in the second equation:
Take the expression for [tex]\(y\)[/tex] from the first equation and substitute it into the second equation:
[tex]\[ 4x + 3y = 14 \][/tex]
[tex]\[ 4x + 3(6 - 2x) = 14 \][/tex]
3. Simplify the second equation:
Distribute the 3:
[tex]\[ 4x + 18 - 6x = 14 \][/tex]
Combine like terms:
[tex]\[ 4x - 6x + 18 = 14 \][/tex]
[tex]\[ -2x + 18 = 14 \][/tex]
4. Solve for [tex]\(x\)[/tex]:
Subtract 18 from both sides to isolate the term with [tex]\(x\)[/tex]:
[tex]\[ -2x = 14 - 18 \][/tex]
[tex]\[ -2x = -4 \][/tex]
Divide both sides by -2:
[tex]\[ x = 2 \][/tex]
5. Substitute [tex]\(x\)[/tex] back into the expression for [tex]\(y\)[/tex]:
Use [tex]\( x = 2 \)[/tex] in the equation [tex]\( y = 6 - 2x \)[/tex]:
[tex]\[ y = 6 - 2(2) \][/tex]
[tex]\[ y = 6 - 4 \][/tex]
[tex]\[ y = 2 \][/tex]
Hence, the solution to the system of equations is:
[tex]\[ x = 2 \][/tex]
[tex]\[ y = 2 \][/tex]
The solution to the system of equations [tex]\(2x + y = 6\)[/tex] and [tex]\(4x + 3y = 14\)[/tex] is [tex]\( x = 2 \)[/tex] and [tex]\( y = 2 \)[/tex].
