Math Problem Statement
Solve the system of linear equations using elimination: 5x - 4y = 19 and 4x + 3y = 9.
Solution
The system of equations shown in your image is:
1) \quad 5x - 4y &= 19 \\ 2) \quad 4x + 3y &= 9 \end{aligned}$$ We will solve this using the elimination method. Here are the steps: ### Step 1: Multiply both equations to make the coefficients of one variable opposites. Let's eliminate $$x$$. To do that, we need to equalize the coefficients of $$x$$. We can multiply the first equation by 4 and the second equation by 5: $$\text{Multiply Eq (1) by 4: } 4(5x - 4y) = 4(19) \implies 20x - 16y = 76$$ $$\text{Multiply Eq (2) by 5: } 5(4x + 3y) = 5(9) \implies 20x + 15y = 45$$ ### Step 2: Subtract the two equations. Now, subtract the second equation from the first to eliminate $$x$$: $$(20x - 16y) - (20x + 15y) = 76 - 45$$ $$20x - 20x - 16y - 15y = 31$$ $$-31y = 31$$ ### Step 3: Solve for $$y$$. $$y = \frac{31}{-31} = -1$$ ### Step 4: Substitute $$y = -1$$ into one of the original equations. Let's substitute $$y = -1$$ into the first equation $$5x - 4y = 19$$: $$5x - 4(-1) = 19$$ $$5x + 4 = 19$$ $$5x = 15$$ $$x = \frac{15}{5} = 3$$ ### Final Solution: The solution to the system is: $$(x, y) = (3, -1)$$ Let me know if you'd like any more details or if you have any questions! --- Here are 5 related questions to expand your understanding: 1. How do you solve a system of equations using substitution instead of elimination? 2. Can the elimination method always be applied to a system of two equations? 3. What happens if the two lines represented by the equations are parallel? 4. How can you verify if your solution to a system of equations is correct? 5. What other methods are used to solve systems of equations? **Tip:** When using the elimination method, always check for the simplest variable to eliminate first to save time.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Linear Equations
System of Equations
Elimination Method
Formulas
Equation of a line: Ax + By = C
Subtraction of equations to eliminate variables
Theorems
Solution to a system of linear equations by elimination
Suitable Grade Level
Grades 8-10
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