Problem 98 - Algebra and Trigonometry

Question

Write a system of equations having {(-2, 7)} as a solution set. (More than one system is possible.)

Step-by-Step Solution

Answer

One possible system of equations having {(-2, 7)} as a solution set is: \[ \begin{cases} y = 2x + 11 \ y = -3x + 1 \end{cases}\]

Step 1: Formulate First Equation

Choose arbitrarely values for \(m\) and \(b\) so that \((-2, 7)\) holds true in the equation \(y = mx + b\). Let the first equation be: \( y = 2x + 11 \). Here, if you substitute \(x = -2\) and \(y = 7\), the equation still holds true.

Step 2: Formulate Second Equation

Similarly, let's formulate another equation. Let this one be: \( y = -3x + 1 \). Here also, if you substitute \(x = -2\) and \(y = 7\), the equation holds true.

Step 3: Write out the System of Equations

The system of equations that satisfies the condition \((-2, 7)\) being a solution is the combination of these two equations:\[ \begin{cases} y = 2x + 11 \ y = -3x + 1 \end{cases}\]

Key Concepts

Solving Linear Systems

Linear systems consist of two or more linear equations involving the same set of variables. The goal in solving such systems is to find the values of the variables that satisfy all equations simultaneously.

Solving a linear system can be approached through various methods, with the aim being to arrive at a solution set represented by the point \( (x,y) \) which satisfies each equation. For example, the point \( (-2, 7) \) will satisfy a system if, when the values \( x=-2 \) and \( y=7 \) are substituted into each equation, they create true statements.

The system of equations can be represented graphically as lines on a coordinate plane, where the solution set is the point or points where the lines intersect. The importance of such a skill cannot be overstated; it forms the foundation for algebra and further mathematics study, including disciplines such as economics, engineering, and the physical sciences.

Substitution Method

The substitution method is one of the algebraic techniques used to solve a system of linear equations. This method involves solving one of the equations for one variable and then 'substituting' this expression into the other equation.

Consider the equations given in the original exercise, \( y = 2x + 11 \) and \( y = -3x + 1 \). As both equations are already solved for \( y \) this step is already done for us. The next step would be to set them equal to each other since they both equal \( y \) and solve for \( x \):

• Set \( 2x + 11 = -3x + 1 \).
• Solve for \( x \) by rearranging terms.

Once \( x \) is determined, we substitute that value back into one of the original equations to find \( y \) and complete the solving process. By doing this, you can arrive at the solution that the system has in common without needing to graph the equations, which proves particularly handy when dealing with numbers that do not easily graph or when precision is crucial.

Algebraic Solutions

Algebraic solutions refer to finding the answer to an equation or system of equations through algebraic manipulations and problem-solving techniques rather than relying on visual models such as graphs. These solutions involve operations such as addition, subtraction, multiplication, division, and substitution to rearrange and solve equations.

Returning to the example from our exercise, when \( (-2, 7) \) is a solution to both \( y = 2x + 11 \) and \( y = -3x + 1 \) separately, it indicates that these linear equations intersect at this point, and algebraic operations can verify that.

Algebraic solutions are fundamental because they allow for precise calculations and are crucial when dealing with abstract problems or models where graphing may be impractical or impossible. Mastery of these skills is critical for students' success in various fields requiring analytical and critical thinking.