Solve the system of equations:
[tex]\left\{\begin{array}{l}
y=-2 x+4 \
y=-3 x+5\n\end{array}\right.[/tex]
Solution: ( $\square$ , $\square$ )
Answer (2)
The solution to the system of equations is (1, 2) by solving for x and then y after setting the two equations equal to each other. First, we found x = 1 and then substituted to find y = 2.
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Set the two equations equal to each other: − 2 x + 4 = − 3 x + 5 .
Solve for x : x = 1 .
Substitute the value of x into one of the original equations to solve for y : y = − 2 ( 1 ) + 4 = 2 .
The solution to the system of equations is ( 1 , 2 ) .
Explanation
Analyze the problem We are given a system of two linear equations:
y = − 2 x + 4 y = − 3 x + 5
Our goal is to find the values of x and y that satisfy both equations.
Set equations equal Since both equations are solved for y , we can set them equal to each other:
− 2 x + 4 = − 3 x + 5
Now, we will solve for x .
Solve for x Add 3 x to both sides of the equation:
− 2 x + 3 x + 4 = − 3 x + 3 x + 5 x + 4 = 5 Subtract 4 from both sides:
x + 4 − 4 = 5 − 4 x = 1 So, x = 1 .
Solve for y Now that we have the value of x , we can substitute it into either of the original equations to find the value of y . Let's use the first equation:
y = − 2 x + 4 Substitute x = 1 :
y = − 2 ( 1 ) + 4 y = − 2 + 4 y = 2 So, y = 2 .
State the solution Therefore, the solution to the system of equations is x = 1 and y = 2 . We can write this as an ordered pair ( 1 , 2 ) .
Examples
Systems of equations are used in many real-world applications, 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, you can set up a system of equations to find the number of units they need to sell to cover their costs. Also, systems of equations are used in physics to solve problems involving multiple forces or velocities. Understanding how to solve systems of equations is a valuable skill in many fields.
