Problem 22 - Contemporary Precalculus

Question

In Exercises \(21-36,\) solve the system. $$\begin{aligned} &2 x+y=-1\\\ &x-3 y=5\\\ &3 x+5 y=-7 \end{aligned}$$

Step-by-Step Solution

Answer

Short Answer: The unique solution for the system of equations \((x,y)=(\frac{2}{7},-\frac{11}{7})\).

Step 1: Solve equation 1 for y

To eliminate one of the variables, solve the first equation for \(y\) in terms of \(x\). We can do this by subtracting \(2x\) from both sides of equation 1: \(y=-1-2x\).

Step 2: Substitute equation 1 for y in equations 2 and 3

Now that we have \(y\) in terms of \(x\), we can substitute it into the other two equations. This will give us two new equations only in terms of \(x\). In equation 2, substitute \(y\) with the expression we just found: \(x-3(-1-2x)=5\). In equation 3, substitute \(y\) with the same expression: \(3x+5(-1-2x)= - 7\).

Step 3: Simplify equations 2 and 3

Simplify both equations by multiplying and simplifying: For equation 2: \(x+3+6x=5 \Rightarrow 7x=2 \Rightarrow x= \frac{2}{7}\). For equation 3: \(3x-5-10x=-7 \Rightarrow -7x=-2 \Rightarrow x= \frac{2}{7}\).

Step 4: Plug x back into y-equation

Now that we have found \(x=\frac{2}{7}\), we can plug this value back into the equation we found for \(y\) in terms of \(x\): \(y=-1-2(\frac{2}{7})\).

Step 5: Solve for y

Finish solving for \(y\) by performing the operations: \(y=-1-\frac{4}{7}=-\frac{11}{7}\).

Step 6: Write the solution

Finally, we can write the solution as an ordered pair: \((x,y)=(\frac{2}{7},-\frac{11}{7})\). This is the unique solution for the system of equations, as it satisfies all three original equations.

Key Concepts

Substitution Method

The substitution method is a popular technique for solving systems of linear equations. This method works by solving one of the equations for one variable in terms of the others and then substituting that expression into the remaining equations. This allows us to reduce the number of variables and simplify the problem.

Here's a step-by-step of how it works:

• First, solve one of the equations for one of the variables.
• Substitute this expression into the other equations.
• Once substituted, you will find that you have equations with fewer variables to solve.
• Continue simplifying and solving the new equations until all variables are determined.

In our example, we first solved for \(y\) in terms of \(x\) from the first equation. Then, replaced \(y\) in the other equations to find \(x\). Finally, once \(x\) was determined, we used it to find \(y\). This method systematically breaks down a problem involving multiple variables into a simpler one.

Simultaneous Equations

Simultaneous equations are a set of equations with multiple variables which all hold true at the same time. Solving them means finding a set of values for the variables that make all the equations true; this is also known as the intersection point.

• Each equation represents a line in a two-dimensional space.
• The solution of a system of simultaneous equations is finding the point where these lines intersect.
• Once the solution is found, it should satisfy all the given equations.

In the given exercise, we had three simultaneous equations with two variables, \(x\) and \(y\). The solution to these equations is the point where all three lines intersect in the plane. The systematic approach of using algebraic manipulation helps us find this intersection point without graphing.

Solution of Equations

The solution to a system of equations is the value or set of values that satisfy every equation in the system. Ensuring accuracy is critical, as any error in calculation can lead to incorrect solutions.

Key points to ensure correct solutions:

• Always check that the calculated values satisfy all the original equations.
• Double-check substitutions and simplifications for any errors.
• If one path to finding the solution seems incorrect or too complex, consider alternative methods like elimination or graphing.

In our exercise, by substituting values back into the original equations after calculating, we verified that \((x,y) = \left(\frac{2}{7}, -\frac{11}{7}\right)\) indeed satisfies all given equations. This final verification step is crucial to ensure that the answer is comprehensive and accurate.