Solve the system. $\left\{\begin{array}{l}x+12 y=68 \\ x=8 y-12\end{array}\right.$
Answer (1)
Substitute the second equation into the first.
Simplify and solve for y : y = 4 .
Substitute y = 4 back into the second equation to find x : x = 20 .
The solution to the system of equations is ( 20 , 4 ) .
Explanation
Understanding the Problem We are given a system of two linear equations with two variables, x and y. The equations are: x + 12y = 68 and x = 8y - 12. The objective is to find the values of x and y that satisfy both equations simultaneously.
Substitution Substitute the second equation (x = 8y - 12) into the first equation (x + 12y = 68). This will result in an equation with only one variable, y: (8y - 12) + 12y = 68.
Solving for y Simplify the equation and solve for y:
Combine like terms: 8 y + 12 y − 12 = 68 20 y − 12 = 68 Add 12 to both sides: 20 y = 68 + 12 20 y = 80 Divide by 20: y = 20 80 y = 4
Solving for x Substitute the value of y (y = 4) back into either of the original equations to solve for x. Using the second equation: x = 8(4) - 12.
Calculating x Calculate the value of x: x = 8 ( 4 ) − 12 x = 32 − 12 x = 20
Final Solution The solution to the system of equations is x = 20 and y = 4. Express the solution as an ordered pair (x, y) = (20, 4).
Examples
Systems of equations are used in various real-world applications, such as determining the break-even point for a business, calculating the optimal mix of ingredients in a recipe, or modeling supply and demand in economics. Understanding how to solve systems of equations helps in making informed decisions and predictions in these scenarios. For example, if you're running a business, you might use a system of equations to figure out how many products you need to sell to cover your costs and start making a profit.
