Math Problem Statement

Solve the following simultaneous equations: 3c + 4d = 10 and 4c + 5d = 13.

Solution

The image shows a system of simultaneous linear equations that needs to be solved. The system is:

3c + 4d &= 10 \\ 4c + 5d &= 13 \end{aligned}$$ We will solve this system using the method of elimination. Here's how we proceed: ### Step 1: Eliminate one variable Multiply both equations to align the coefficients of one of the variables. Let's eliminate $$c$$. We multiply the first equation by 4 and the second equation by 3 to make the coefficients of $$c$$ in both equations equal: $$\begin{aligned} 4(3c + 4d) &= 4(10) \quad \Rightarrow \quad 12c + 16d = 40 \\ 3(4c + 5d) &= 3(13) \quad \Rightarrow \quad 12c + 15d = 39 \end{aligned}$$ ### Step 2: Subtract the equations Now, subtract the second equation from the first to eliminate $$c$$: $$(12c + 16d) - (12c + 15d) = 40 - 39$$ Simplifying: $$d = 1$$ ### Step 3: Solve for $$c$$ Substitute $$d = 1$$ into one of the original equations, say $$3c + 4d = 10$$: $$3c + 4(1) = 10 \quad \Rightarrow \quad 3c + 4 = 10$$ Solving for $$c$$: $$3c = 6 \quad \Rightarrow \quad c = 2$$ ### Final Solution: $$c = 2, \quad d = 1$$ Would you like more details or have any questions about the solution? --- Here are 5 related questions to explore further: 1. How can you solve a system of equations using substitution instead of elimination? 2. What are the conditions under which a system of linear equations has no solution? 3. How do you determine if a system of equations has infinitely many solutions? 4. Can you solve a system of three linear equations using elimination? How does the process differ? 5. What is the graphical interpretation of solving two simultaneous linear equations? **Tip:** When using elimination, it's useful to align the coefficients of one variable by multiplying the equations before subtracting or adding them.

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Math Problem Analysis

Mathematical Concepts

Simultaneous Equations
Linear Equations
Elimination Method

Formulas

Equation 1: 3c + 4d = 10
Equation 2: 4c + 5d = 13

Theorems

Elimination method for solving simultaneous equations

Suitable Grade Level

Grades 9-10

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