Math Problem Statement
In order to create a system of equations that has the same solution as the system { x + y = 4, 2x + y = 7 } complete three steps. Step 1: Multiply the equation x + y = 4 by ______. Step 2: Add the equations 2x + y = 7 and ______. Step 3: Write the new system of equations as ______.
Solution
The two images you provided are part of a system of equations problem. Based on the first image, you want to create a system of equations that has the same solution as the following system:
x + y &= 4 \\ 2x + y &= 7 \end{aligned}$$ Let's break down the steps as shown in the second image: 1. **Step 1**: Multiply the equation $$x + y = 4$$ by **$$-2$$** to eliminate $$x$$ in the system. $$-2(x + y) = -2(4) \implies -2x - 2y = -8$$ 2. **Step 2**: Add the equation $$2x + y = 7$$ to the equation $$-2x - 2y = -8$$. $$(-2x - 2y) + (2x + y) = -8 + 7 \implies -y = -1$$ 3. **Step 3**: The new system of equations after these steps would be: $$\begin{aligned} x + y &= 4 \\ -y &= -1 \end{aligned}$$ Thus, the correct choices for completing the sentences are: - **Step 1**: Multiply by **$$-2$$** to get the equation $$-2x - 2y = 4$$. - **Step 2**: Add the equations $$2x + y = 7$$ and $$-2x - 2y = -8$$. - **Step 3**: The new system is $$x + y = 4$$ and $$-y = -1$$. Would you like more details or help with further steps? Let me know if you have any questions. Here are some related questions you could explore next: 1. How do you solve the system of equations $$x + y = 4$$ and $$-y = -1$$? 2. What methods can you use to solve a system of linear equations? 3. How can you check the solutions to a system of equations? 4. What would happen if you multiplied the second equation by a different number instead of $$-2$$? 5. How do you graphically represent systems of equations? Tip: Always check for consistency in the system after transforming equations; sometimes it's easy to lose track of the solution as you manipulate them!Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
System of Equations
Algebra
Linear Equations
Elimination Method
Formulas
x + y = 4
2x + y = 7
Theorems
Elimination Method
Suitable Grade Level
Grades 8-9
Related Recommendation
Solving Systems of Equations Using Elimination
Using the Elimination Method to Solve Systems of Equations
Solving Linear Equations by Elimination Method
Elimination of x in Linear Equations: 3x + 2y = 65 and x - 5y = -1
Solving a System of Equations Using Elimination Method: 4x + 9y = 21 and 2x - 2y = 4