Problem 16 - Thinking Mathematically

Question

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

Step-by-Step Solution

Answer

The solution to the system of equations \(\left\{\begin{array}{l}2 x-3 y=-13 \ y=2 x+7\end{array}\right.\) is \(x = 2\) and \(y = 11\).

Step 1: Substitute \(y\) from the second equation into first equation

We can take \(y = 2x + 7\) from the second equation and substitute it into the first equation: \n\(2x - 3(2x + 7) = -13\)

Step 2: Simplify the equation to solve for \(x\)

Expand and simplify the equation to isolate \(x\): \n\(2x - 6x - 21 = -13\) \nSolving -4x - 21 = -13 gives \(x = 2\)

Step 3: Substitute \(x\) into second equation to find \(y\)

Now that we have solved \(x = 2\), substitute it back into the equation \(y = 2x + 7\) to find \(y\): \n\(y = 2*2 + 7, y = 11\)

Step 4: Checking the solution

We can verify that x = 2 and y = 11 are indeed solutions for the system by substituting them into the original equations: \n\(2*2 - 3*11 = -13\) and \n\(11 = 2*2 + 7\) Both equations are satisfied, so the solution is correct

Key Concepts

Substitution Method

When you're solving systems of equations, the substitution method can be a powerful tool. It involves replacing one variable with another to simplify the problem. Imagine you have a pair of shoes tied together – by understanding one, you automatically know where the other one goes. In the exercise, we used this method by taking the expression for 'y' from one equation and plugging it into the other. It's like a magic trick; swap out 'y' for its equivalent '2x+7', and voila, you have an equation with just one variable to solve!

This approach works especially well when one of the equations in the system has already been solved for a single variable. It's like getting a free pass – one step of the work is already done for you! Just make sure to simplify the new equation carefully, solving for the unknown, and then back-substitute to find the second variable's value.

Systems of Equations

A system of equations is essentially a team of algebraic expressions that must work together to find a common solution. In the given example, we have two allies – two linear equations that intersect at a certain point on a graph. This intersection represents the values of 'x' and 'y' that satisfy both equations simultaneously. Systems can have one solution, no solution (parallel lines that never meet), or infinite solutions (the same line, so they meet everywhere).

The beauty of solving these systems algebraically, as compared to graphing, is precision. Algebraic methods allow you to find the exact values without the messiness of drawing lines – neat and tidy!

Algebraic Solutions

Delving into algebraic solutions is the heart of solving systems of equations. It's the analytical pathway to crack the code of the variables' values. By performing operations like substitution, you're essentially conducting a negotiation between the equations, reshuffling the pieces until the picture is clear.

In this example, we initially face a jumble of 'x's and 'y's. However, as we simplify and re-arrange the equations, they start to make sense. Through the course of substitution and simplification, the nebulous cloud of algebra clears, revealing the values of 'x = 2' and 'y = 11'. It's important to do this step by step to minimize errors and turn complex problems into simple, solvable puzzles.

Mathematical Problem-Solving

Mathematical problem-solving is like being a detective. It's about following clues (steps), checking your evidence (solutions), and arriving at a conclusion (the answer that works for all equations). As we followed the clues in this exercise by substituting and solving, we found our suspect, or rather, the solution.

It's crucial to check your work. Once you have your proposed solution, plug it back into the original equations to ensure they make sense. In our case, we checked and confirmed 'x = 2' and 'y = 11' were indeed the culprits that satisfied both equations – case closed! Remember, problem-solving in math is creative, logical, and, above all, methodical.