Problem 8 - College Algebra

Question

Solve each system by the substitution method. \(\left\\{\begin{array}{l}2 x-3 y=-13 \\ y=2 x+7\end{array}\right.\)

Step-by-Step Solution

Answer

The solution to the system of equations will be the values of x and y obtained from the above steps.

Step 1: Express Variable in Terms of the Other

Starting with the linear equation \(3x - y = 5\), express \(y\) in terms of \(x\) which would lead to \(y = 3x - 5.\)

Step 2: Substitute for the Variable in the Other Equation

Substitute the value of \(y\) into the quadratic equation: \(x^{2} + (3x - 5)^{2} - 5.\)

Step 3: Simplify and Solve

Simplify the expression to form a quadratic equation, \(x^{2} + (9x^{2} - 30x + 25) - 5 = x^{2} + 9x^{2} - 30x + 20 = 10x^{2} - 30x + 20. Solve the quadratic equation to get values for \(x\).

Step 4: Substitute Back

Substitute \(x\) back into the equation \(y = 3x - 5\) to get the values of \(y\)

Key Concepts

Linear Equations

Linear equations are fundamental in algebra and involve expressions where variables are only raised to the first power. These are equations of the form \(ax + b = c\) where \(a\), \(b\), and \(c\) are constants. They graph as straight lines on a coordinate plane.

• In the context of solving systems by substitution, one linear equation can often be used to express one variable in terms of another, simplifying a potentially complex problem.
• For instance, given the equation \(3x - y = 5\), we solve for \(y\) and get \(y = 3x - 5\). This equation is now a simple expression ready to substitute into another equation.

Solving linear equations involves finding the value of the variable that makes the equation true. It is about understanding the relationship between the components within the equation to isolate the unknowns effectively. Breaking the problem into manageable steps, much like a puzzle, allows you to see how changes in one variable affect another.

Quadratic Equations

Quadratic equations are expressions where the variable is raised to the second power, typically written in the form \(ax^2 + bx + c = 0\). The solutions to quadratic equations give rise to parabolic graphs on a coordinate plane.

• When substituting into a quadratic equation, as in our exercise \(x^2 + (3x - 5)^2 - 5\), it is crucial to perform algebraic operations carefully to ensure accuracy.
• The expanded form \((3x - 5)^2\) translates to \((9x^2 - 30x + 25)\), effectively showing how each term interacts when squared.

Solving these equations often requires factoring, completing the square, or using the quadratic formula. Quadratic equations involve deeper algebraic manipulation, including checking your work with one or more solution methods to confirm accuracy. Understanding how they differ from linear equations in complexity and in graphical representation is key in mastering systems of equations.

Systems of Equations

Systems of equations involve finding solutions that satisfy all equations within the system simultaneously. These can be composed of both linear and non-linear equations, such as quadratic equations.

• The substitution method is a popular choice for solving systems because it transforms the problem into a manageable one by handling one equation at a time.
• The process involves expressing one of the variables from an equation and substituting it into another equation, as seen by substituting \(y = 3x - 5\) into the quadratic equation.

Understanding how to approach systems requires a strategic mindset. Rather than tackling all equations at once, the substitution method helps in simplifying the problem by reducing it to simpler parts. Moreover, it demonstrates the mathematician's toolkit of isolating variables to see interactions across different equations. This step-by-step approach not only aids in gaining solutions but also enhances logical reasoning skills in mathematical contexts.