Solve the system of equations using the substitution method. The value of $y$ is $\qquad$.
$\begin{array}{l}
-2 x-y=-3 \\
y=-3 x+2
\end{array}$
The solution is $\square$
Answer (1)
Substitute the expression for y from the second equation into the first equation: − 2 x − ( − 3 x + 2 ) = − 3 .
Simplify and solve for x : x = − 1 .
Substitute the value of x back into the equation y = − 3 x + 2 to find y : y = 5 .
The value of y is 5 .
Explanation
Analyze the problem We are given a system of two equations with two variables, x and y . Our goal is to solve for the value of y using the substitution method. The given equations are:
− 2 x − y = − 3 y = − 3 x + 2
The substitution method involves solving one equation for one variable and substituting that expression into the other equation. In this case, the second equation is already solved for y , so we can substitute the expression − 3 x + 2 for y in the first equation.
Substitute and solve for x Substitute the expression for y from the second equation into the first equation:
− 2 x − ( − 3 x + 2 ) = − 3
Now, simplify the equation and solve for x :
− 2 x + 3 x − 2 = − 3 x − 2 = − 3 x = − 3 + 2 x = − 1
Solve for y Now that we have the value of x , we can substitute it back into the equation y = − 3 x + 2 to find the value of y :
y = − 3 ( − 1 ) + 2 y = 3 + 2 y = 5
Final Answer Therefore, the value of y is 5.
Examples
Substitution method is a fundamental technique in algebra, useful in various real-world scenarios. For instance, consider a business where the cost C of producing x items is given by C = 2 x + y , and the selling price y of each item depends on the number of items produced as y = 5 x + 10 . By substituting the expression for y into the cost equation, the business can determine the cost C solely in terms of the number of items produced x , helping in cost analysis and production planning. This method simplifies complex relationships, making it easier to understand and optimize business operations.
