solve using substitution.

Question

y = 2x + 3
3x + 2y = 12
(give details)

Answer 1

To solve the system of equations using substitution, we can start with the two equations given:

1. $y=2x+3$ (Equation 1)
2. $3x+2y=12$ (Equation 2)

Step 1: Substitute $y$ in Equation 2

Since Equation 1 gives us $y$ in terms of $x$, we can substitute this expression for $y$ into Equation 2.

Substituting $y$ in Equation 2: $$3x+2(2x+3)=12$$

Step 2: Simplify the equation

Now distribute $2$ to both terms in the parentheses: $$3x+4x+6=12$$

Combine like terms: $$7x+6=12$$

Step 3: Solve for $x$

Subtract $6$ from both sides: $$7x=12-6$$ $$7x=6$$

Now, divide by $7$: $$x=\frac{6}{7}$$

Step 4: Substitute back to find $y$

Now that we have $x$, we substitute it back into Equation 1 to find $y$: $$y=2\left(\frac{6}{7}\right)+3$$

Calculate $y$: $$y=\frac{12}{7}+3$$ To combine these, convert $3$ to a fraction with a denominator of $7$: $$y=\frac{12}{7}+\frac{21}{7}=\frac{12+21}{7}=\frac{33}{7}$$

Final answer

The solution to the system of equations is: $$x=\frac{6}{7}\text{and}y=\frac{33}{7}$$

Thus, the solution as an ordered pair is: $$(\frac{6}{7},\frac{33}{7})$$