Let's solve the system of linear equations step by step:
Given:
[tex]\[ 2x + y = 6 \][/tex]
[tex]\[ 4x + 3y = 14 \][/tex]
Step 1: Solve one equation for one variable.
Let's solve the first equation for [tex]\( y \)[/tex]:
[tex]\[ 2x + y = 6 \][/tex]
[tex]\[ y = 6 - 2x \][/tex]
Step 2: Substitute this expression into the second equation.
Substitute [tex]\( y = 6 - 2x \)[/tex] into the second equation:
[tex]\[ 4x + 3(6 - 2x) = 14 \][/tex]
Step 3: Simplify and solve for [tex]\( x \)[/tex].
Expand and simplify the equation:
[tex]\[ 4x + 18 - 6x = 14 \][/tex]
Combine like terms:
[tex]\[ -2x + 18 = 14 \][/tex]
Subtract 18 from both sides:
[tex]\[ -2x = -4 \][/tex]
Divide by -2:
[tex]\[ x = 2 \][/tex]
Step 4: Substitute [tex]\( x = 2 \)[/tex] back into the expression for [tex]\( y \)[/tex].
Substitute [tex]\( x = 2 \)[/tex] into the expression [tex]\( y = 6 - 2x \)[/tex]:
[tex]\[ y = 6 - 2(2) \][/tex]
[tex]\[ y = 6 - 4 \][/tex]
[tex]\[ y = 2 \][/tex]
Conclusion:
The solutions to the system of equations are:
[tex]\[ x = 2 \][/tex]
[tex]\[ y = 2 \][/tex]
Thus, the solution to the system of equations [tex]\(2x + y = 6\)[/tex] and [tex]\(4x + 3y = 14\)[/tex] is [tex]\((x, y) = (2, 2)\)[/tex].
