Problem 19 - Thinking Mathematically

Question

Solve each system by the substitution method. Be sure to check all proposed solutions. \(\left\\{\begin{array}{l}2 x-y=-5 \\ x+5 y=14\end{array}\right.\)

Step-by-Step Solution

Answer

The solution to the system of equations is \(x = -1\) and \(y = 3\).

Step 1: Arrange An Equation

The first step in the substitution method is to solve one of the equations for one variable in terms of the other variable. The second equation, \(x+5y=14\), is more convenient for this purpose. It can be rearranged to form \(x=14-5y\). This represents \(x\) in terms of \(y\).

Step 2: Substitute into Other Equation

Substitute \(x = 14 - 5y\) into the first equation. So, \(2(14-5y) - y = -5\). This reduces to \(28 - 10y - y = -5\), which simplifies to \(28 - 11y = -5\).

Step 3: Solve For Variable

This equation can be solved for \(y\). First, add \(11y\) on both sides, resulting in \(28 = 11y - 5\). Then, add 5 to both sides, which gives \(33 = 11y\). Dividing both sides by 11 yields \(y = 3\).

Step 4: Find Other Variable

Substitute \(y = 3\) into the equation \(x = 14 - 5y\), which gives \(x = 14 - 5(3) = -1\).

Step 5: Check Proposed Solution

Check the solution \((-1,3)\) by substituting into both original equations to see if both are true. For the first equation \(2x - y = -5\), substituting \(x = -1\) and \(y = 3\) results in \(-2*1 - 3 = -5\), which is true. For the second equation \(x + 5y = 14\), substituting \(x = -1\) and \(y = 3\) gives \(-1 + 5*3 = 14\), which is also true. Thus, the solution checks out.

Key Concepts

system of equations

A system of equations consists of two or more linear equations that work hand-in-hand to explain relationships between variables. In this scenario, the equations are \(2x - y = -5\) and \(x + 5y = 14\). Each equation represents a line on a graph. The solution to the system is the point where these lines meet, showing the values of \(x\) and \(y\) that satisfy both equations simultaneously.
Understanding how these systems work is crucial. Each equation is like a rule or a condition. We need to find an \(x\) and a \(y\) that obey both rules at the same time. This is why the two lines intersect at a specific point for linear systems. That point is the solution!
By using the substitution method, we focus on expressing one variable in terms of the other using one equation, making it easier to find this common point where both conditions are met.

solving linear equations

Solving linear equations is all about finding the value of the variables, typically \(x\) and \(y\), that fulfill the given equations' requirements. With substitution, the aim is to express one variable in terms of another, which simplifies the problem.
Let's go through the process:

• Start by selecting one equation and solving it for one variable. In our case, the second equation \(x + 5y = 14\) is rearranged as \(x = 14 - 5y\).
• This expression for \(x\) is then substituted into the other equation. For this system, replace \(x\) in \(2x - y = -5\) with \(14 - 5y\).
• Now, you'll have a single equation in terms of \(y\): like \(28 - 11y = -5\). Solve for \(y\) to find its value.
• Once \(y\) is known, insert it back into the expression for \(x\) to find \(x\).

Following these steps allows us to break down the complexity of handling two equations into more manageable parts. This builds a pathway towards retrieving the solution for the system.

checking solutions

Checking solutions is an essential part of solving systems of equations. Once you find potential solutions like \((-1, 3)\), it's critical to verify they genuinely satisfy all original conditions. This verifies the accuracy of your calculations.
Here's how:

• Take the found solution \(x=-1\) and \(y=3\) and substitute them back into the original equations.
• For the first equation, \(2x - y = -5\), replacing gives \(2(-1) - 3 = -5\). Check if this equation holds true.
• Then, try the second equation \(x + 5y = 14\). Substitute, getting \(-1 + 5(3) = 14\). Ensure this makes sense too.

If both checks are successful, congratulations! This means that the pair \((-1, 3)\) is indeed the correct solution and fits perfectly at the intersection where the two lines meet. Always remember, checking your answers anchors your understanding and ensures your solution's correctness.