\begin{array}{c}
3 x-2 y=-12 \
y=4 x+1
\end{array}
Answer (2)
To solve the system of equations 3 x − 2 y = − 12 and y = 4 x + 1 , we substituted and simplified to find x = 2 and y = 9 . The final solution is the ordered pair ( 2 , 9 ) .
;
Substitute the expression for y from the second equation into the first equation.
Simplify and solve for x : 3 x − 2 ( 4 x + 1 ) = − 12 ⇒ x = 2 .
Substitute the value of x back into the second equation to find y : y = 4 ( 2 ) + 1 = 9 .
The solution to the system of equations is ( 2 , 9 ) .
Explanation
Understanding the Problem We are given a system of two equations with two variables, x and y:
Equation 1: 3 x − 2 y = − 12
Equation 2: y = 4 x + 1
Our objective is to solve this system of equations to find the values of x and y. We will use the substitution method.
Substitution Substitute the expression for y from Equation 2 into Equation 1:
3 x − 2 ( 4 x + 1 ) = − 12
Simplifying the Equation Expand and simplify the equation:
3 x − 8 x − 2 = − 12
− 5 x − 2 = − 12
Isolating x Add 2 to both sides of the equation:
− 5 x = − 12 + 2
− 5 x = − 10
Solving for x Divide both sides by -5 to solve for x:
x = − 5 − 10
x = 2
Solving for y Substitute the value of x back into Equation 2 to find the value of y:
y = 4 ( 2 ) + 1
y = 8 + 1
y = 9
Final Answer The solution to the system of equations is the ordered pair (x, y) = (2, 9).
Examples
Systems of equations are used in various real-life situations, such as determining the break-even point for a business. For example, if a company has fixed costs and variable costs, and they sell a product at a certain price, we can set up a system of equations to find the number of units they need to sell to cover their costs. Similarly, in physics, systems of equations can be used to analyze the forces acting on an object in equilibrium.
