Solve the system. $\left\{\begin{array}{l}y=2 x-1 \\ 3 x-y=-1\end{array}\right.
Answer (2)
Substitute the first equation into the second equation.
Solve for x : x = − 2 .
Substitute the value of x back into the first equation to find y : y = − 5 .
The solution to the system of equations is ( − 2 , − 5 ) .
Explanation
Analyze the problem We are given a system of two linear equations:
y = 2 x − 1 3 x − y = − 1
Our goal is to find the values of x and y that satisfy both equations simultaneously. We can use the substitution method to solve this system.
Solve for x Substitute the first equation into the second equation to eliminate y :
3 x − ( 2 x − 1 ) = − 1
Simplify the equation:
3 x − 2 x + 1 = − 1 x + 1 = − 1
Solve for x :
x = − 1 − 1 x = − 2
Solve for y Now that we have the value of x , we can substitute it back into the first equation to find the value of y :
y = 2 ( − 2 ) − 1 y = − 4 − 1 y = − 5
State the solution So the solution to the system of equations is x = − 2 and y = − 5 . This corresponds to the point ( − 2 , − 5 ) .
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. For example, suppose a company wants to know how many units of a product they need to sell to cover their fixed costs and variable costs. By setting up a system of equations representing the cost and revenue functions, they can find the break-even point where cost equals revenue. This helps them make informed decisions about pricing, production, and sales targets.
The solution to the system of equations is (-2, -5), found using substitution to solve for x and y. After substituting y from the first equation into the second, we simplified and solved for x, then substituted back to find y. Thus, the point (-2, -5) is where both equations intersect.
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