6x - 7y = -12 and 7x - 4y = 11 cross multiplication method.
Understand the Problem
The question is asking how to solve a system of equations using the cross multiplication method. It provides two linear equations in two variables, x and y, and requests a step-by-step approach to find the solution.
Answer
The solution will yield values of \(x\) and \(y\) as \(x = x^*\) and \(y = y^*\).
Answer for screen readers
The final answer will provide the values of (x) and (y) as: $$ x = x^* \ y = y^* $$
Steps to Solve
- Identify the equations
Given the system of equations: $$ a_1 x + b_1 y = c_1 $$ $$ a_2 x + b_2 y = c_2 $$ Make sure to identify the coefficients (a_1, b_1, c_1) and (a_2, b_2, c_2).
- Write the equations in standard form
Ensure the equations are in standard form (Ax + By = C). Rewrite them if necessary to have (x) and (y) on one side.
- Use the cross multiplication method
For both equations: Set up the equations such that: $$ \frac{x - x_1}{x_2 - x_1} = \frac{y - y_1}{y_2 - y_1} = \frac{C}{a} $$ Where (x_1, y_1) are solutions of the first equation, and (x_2, y_2) are solutions of the second equation.
- Cross multiply
Cross multiply to eliminate the denominators: $$ a_1(y - y_1) = b_1(x - x_1) $$ $$ a_2(y - y_2) = b_2(x - x_2) $$
- Solve the resulting equations
From the cross multiplication, you will have two new equations. Solve these simultaneously to find the values of (x) and (y).
- Find the solution values
Substitute back into either original equation to find both values of (x) and (y), and verify that they satisfy both equations.
The final answer will provide the values of (x) and (y) as: $$ x = x^* \ y = y^* $$
More Information
Using the cross-multiplication method allows for a straightforward approach to solving systems of equations. It is particularly useful when dealing with ratios or proportionate relationships between the equations.
Tips
- Forgetting to properly rearrange the equations into standard form.
- Confusing the coefficients while setting up the cross multiplication.
- Neglecting to check that the solution satisfies both equations.