Question

x+y=4
2x3y=7
solve using substitution

Answer 1

x+y=4
2x+3y=7
solve using substitution

Answer 2

From the first equation, we can solve for y:
y = 4 - x

Substitute this into the second equation:
2x(3(4 - x)) = 7

Simplify:
2x(12 - 3x) = 7
24x - 6x^2 = 7

Rearrange the equation to a quadratic form:
6x^2 - 24x + 7 = 0

Solve for x using the quadratic formula:
x = (-(-24) ± √((-24)^2 - 4(6)(7))) / (2(6))

Simplify:
x = (24 ± √(576 - 168)) / 12
x = (24 ± √408) / 12
x = (24 ± √(4 * 102)) / 12
x = (24 ± 2√102) / 12
x = (2(12 ± √102)) / 12
x = (12 ± √102) / 6

x = (12 + √102) / 6 or x = (12 - √102) / 6

For the first solution, x = (12 + √102) / 6, substitute this back into the equation for y:
y = 4 - ((12 + √102) / 6)

Simplify:
y = 24/6 - (12 + √102) / 6
y = (12 - 12 - √102) / 6
y = -√102 / 6
y = -√102 / 6

The first solution is x = (12 + √102) / 6 and y = -√102 / 6.

For the second solution, x = (12 - √102) / 6, substitute this back into the equation for y:
y = 4 - ((12 - √102) / 6)

Simplify:
y = 24/6 - (12 - √102) / 6
y = (12 - 12 + √102) / 6
y = √102 / 6
y = √102 / 6

The second solution is x = (12 - √102) / 6 and y = √102 / 6.

Therefore, the solutions to the system of equations are (x, y) = ((12 + √102) / 6, -√102 / 6) and ((12 - √102) / 6, √102 / 6).

Answer 3

To solve the system of equations using substitution, we will solve one equation for one variable and then substitute that value into the other equation.

First, let's solve the first equation, x + y = 4, for x:
x = 4 - y

Now, substitute this expression for x into the other equation, 2x + 3y = 7:
2(4 - y) + 3y = 7

Expanding and simplifying:
8 - 2y + 3y = 7
8 + y = 7
y = 7 - 8
y = -1

Now, substitute this value for y back into the first equation to solve for x:
x + (-1) = 4
x - 1 = 4
x = 4 + 1
x = 5

Therefore, the solution to the system of equations x + y = 4 and 2x + 3y = 7 is x = 5 and y = -1.