Math Problem Statement

Solve the following system of equations.

Solve the following system of equations: 6x - 7y = -9 -8x + 9y = 11

Solution

To solve the given system of equations, we can use the method of elimination or substitution. The given system is:

6x - 7y = -9 \quad \text{(1)} \\ -8x + 9y = 11 \quad \text{(2)} \end{cases}$$ Let's use the elimination method. First, we need to eliminate one of the variables. We can do this by making the coefficients of $$x$$ or $$y$$ the same in both equations. Here, we'll eliminate $$x$$. Multiply equation (1) by 8 and equation (2) by 6: $$\begin{cases} 8(6x - 7y) = 8(-9) \\ 6(-8x + 9y) = 6(11) \end{cases}$$ This simplifies to: $$\begin{cases} 48x - 56y = -72 \quad \text{(3)} \\ -48x + 54y = 66 \quad \text{(4)} \end{cases}$$ Now, add equations (3) and (4): $$48x - 56y - 48x + 54y = -72 + 66$$ Simplify the left-hand side and right-hand side: $$-2y = -6$$ Solve for $$y$$: $$y = 3$$ Now that we have $$y = 3$$, substitute $$y$$ back into one of the original equations to solve for $$x$$. We'll use equation (1): $$6x - 7(3) = -9$$ $$6x - 21 = -9$$ Add 21 to both sides: $$6x = 12$$ Divide by 6: $$x = 2$$ Therefore, the solution to the system of equations is: $$x = 2, \; y = 3$$ Would you like more details or have any questions? Here are some related questions for further practice: 1. Solve the system of equations using the substitution method: $$\begin{cases} 4x + 5y = 2 \\ 3x - y = 7 \end{cases}$$ 2. Determine if the following system of equations is consistent, inconsistent, or dependent: $$\begin{cases} x - 2y = 3 \\ 2x - 4y = 6 \end{cases}$$ 3. Solve the system of linear equations by graphing: $$\begin{cases} y = 2x - 1 \\ y = -x + 3 \end{cases}$$ 4. Solve the system of equations using matrix methods: $$\begin{cases} 2x + 3y = 8 \\ 5x - y = 7 \end{cases}$$ 5. Use the elimination method to solve the following system: $$\begin{cases} 3x + 2y = 16 \\ 4x - 5y = -9 \end{cases}$$ **Tip:** When solving systems of equations, always double-check your solution by substituting the values of the variables back into the original equations.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Systems of Equations

Formulas

Elimination method for solving systems of linear equations

Theorems

Addition and elimination method for solving linear equations

Suitable Grade Level

Grades 9-11

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