Question

Solve the system \(\left\\{\begin{array}{l}3 x+2 y=5 \\ 7 x+5 y=1\end{array}\right.\) by changing variables \(\left\\{\begin{array}{l}x=5 x^{\prime}-2 y^{\prime} \\ y=-7 x^{\prime}+3 y^{\prime}\end{array}\right.\) and solving the re- sulting equations for \(x^{\prime}\) and \(y^{\prime}\).

Short answer

\(x' = 5\) and \(y' = 1\).

Step-by-step solution

Substitute the Variable Change

Substitute the given change of variables into the first equation of the system: \(3x + 2y = 5\). Using \(x = 5x' - 2y'\) and \(y = -7x' + 3y'\), we have:\[3(5x' - 2y') + 2(-7x' + 3y') = 5\].

Simplify the First Equation

Expanding the substituted equation, we get:\[15x' - 6y' - 14x' + 6y' = 5\].Simplify the equation by combining like terms:\[x' = 5\].

Substitute into the Second Original Equation

Substitute the change of variables into the second equation: \(7x + 5y = 1\). Using \(x = 5x' - 2y'\) and \(y = -7x' + 3y'\), we have:\[7(5x' - 2y') + 5(-7x' + 3y') = 1\].

Simplify the Second Equation

Expand the expression:\[35x' - 14y' - 35x' + 15y' = 1\].Combine like terms:\[y' = 1\].

Solve for \(x'\) and \(y'\)

From Step 2, we determined \(x' = 5\) and from Step 4, \(y' = 1\). Thus, the solutions for the new variables are \(x' = 5\) and \(y' = 1\).

Final Step: Interpretation

The solution of the given system of equations in terms of \(x'\) and \(y'\) is \(x' = 5\) and \(y' = 1\). These values satisfy both transformed equations with the substituted variable change.

Key concepts

Substitution Method

The substitution method is a technique often employed when solving systems of linear equations. It involves solving one equation for a single variable and then substituting this expression into another equation. This method effectively reduces the number of equations in a system_{making it more manageable}.

For example, consider solving the system of equations given in the exercise: the method starts by isolating one variable in terms of the other. The exercise involves a system where a transformation is applied, substituting new expressions for the variables. By inserting the expressions for \(x\) and \(y\) from the transformation, you introduce the new variables \(x'\) and \(y'\) into the equations.

In this context, substitution is not only about direct numerical solving; it’s about applying given expressions to transform and simplify equations before reaching the solutions for the new variables.

Variable Transformation

Variable transformation involves changing variables in an equation or a system of equations. In the exercise provided, you have expressions replacing \(x\) and \(y\) with \(5x' - 2y'\) and \(-7x' + 3y'\) respectively.

This technique helps make complex systems more solvable. By transforming variables, you're often able to reduce a complicated system into an easier one to solve. This process can also reveal new insights or properties of the system.

• Start by substituting the transformed variable expressions into each of the original equations.
• Simplify the resulting new equations to isolate \(x'\) and \(y'\).
• Solve the simplified system to find the values of these new variables.

Through this process, you've maintained equivalency with the original system while working in a form that's easier to handle. This is especially useful in linear algebra where transformations can simplify both interpretation and solution.

Linear Algebra

Linear algebra is an area of mathematics focusing on linear equations, linear functions, and their representations through matrices and vector spaces. It provides a framework for understanding systems of equations, like the one in the exercise, using various methods, including substitution and transformation.

When solving systems of equations in linear algebra:

• You may encounter various techniques such as elimination or substitution for reducing complexity.
• Transformations allow simplification and sometimes reveal inherent properties of the system.
• Understanding concepts like matrices and vectors helps interpret solutions and connect different systems to general theories.

In the context of the exercise, after transforming and simplifying the system, the application of linear algebra principles ensures that you arrive at consistent solutions \((x' = 5)\) and \((y' = 1)\) for new variables. This approach is central to many scientific and engineering problems.