Problem 7 - Precalculus

Question

Solve each system of linear equations by substitution. $$\begin{aligned} &2 u+5 v=7\\\ &3 u-v=5 \end{aligned}$$

Step-by-Step Solution

Answer

\(u = \frac{32}{17}\) and \(v = \frac{11}{17}\).

Step 1: Solve one equation for one variable

Start with the second equation: \(3u - v = 5\). Solve for \(v\) by isolating it on one side. First, add \(v\) to both sides to get \(3u = v + 5\), then subtract 5 from both sides to get \(v = 3u - 5\).

Step 2: Substitute the expression into the other equation

Take the expression from Step 1, \(v = 3u - 5\), and substitute \(v\) in the first equation: \(2u + 5(3u - 5) = 7\).

Step 3: Simplify and solve for the first variable

Distribute the 5 in the substituted equation: \(2u + 15u - 25 = 7\). Combine like terms to get \(17u - 25 = 7\). Add 25 to both sides to get \(17u = 32\). Finally, divide both sides by 17 to solve for \(u\): \(u = \frac{32}{17}\).

Step 4: Substitute back to find the second variable

Substitute \(u = \frac{32}{17}\) into the expression for \(v\) from Step 1: \(v = 3\left(\frac{32}{17}\right) - 5\). Evaluate: \(v = \frac{96}{17} - \frac{85}{17}\). Simplify: \(v = \frac{11}{17}\).

Step 5: Verify the solution

Substitute \(u = \frac{32}{17}\) and \(v = \frac{11}{17}\) into the original equations to verify. For the first equation: \(2\left(\frac{32}{17}\right) + 5\left(\frac{11}{17}\right) = \frac{64}{17} + \frac{55}{17} = \frac{119}{17} = 7\). For the second equation: \(3\left(\frac{32}{17}\right) - \frac{11}{17} = \frac{96}{17} - \frac{11}{17} = \frac{85}{17} = 5\). Both equations are satisfied.

Key Concepts

Understanding the Substitution Method

The substitution method is a fundamental technique for solving systems of linear equations. It's particularly useful when one of the equations in your system can be easily solved for one variable in terms of the others. By isolating one variable, you can simplify the system and solve for the variables one at a time.

**Steps to implement substitution:**

• Isolate one variable: Start by manipulating one equation to solve for one of the variables. For instance, let's say we have the equation \(3u - v = 5\). Here, it's efficient to solve for \(v\), yielding \(v = 3u - 5\).
• Substitute into the other equation: Once you have an expression for \(v\), substitute it back into the other equation in the system. This replaces one variable in the second equation, allowing you to solve it with just one unknown.

Substitution is a step-by-step process that reduces the problem to a simpler form, making it easier to find the values of the variables. This method helps keep the equations organized and manageable, especially when dealing with complex systems.

Explaining Linear Equations

Linear equations form the backbone of algebra and are equations of the first degree, meaning the highest power of the variable is one. When graphed, they form straight lines, which is why they're called "linear."

**Parts of a linear equation:**

• Variables: Represent unknown values, commonly shown as \(x\), \(y\), or other letters.
• Constants: Fixed numbers that stand alone or multiply variables.

Linear equations often appear in the form \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. Solving these equations involves finding the values of the variables that make the equation true.

In systems of linear equations, like \(2u + 5v = 7\) and \(3u - v = 5\), you have more than one linear equation, and you need to find values for the variables that satisfy all the equations simultaneously. These can be visualized as lines on a graph, and the solution is the point where the lines intersect.

Verification of Solutions

Verifying solutions is a crucial step in solving systems of equations. It ensures that the values you've computed indeed satisfy the original equations. This step confirms the correctness of your results and is critical, especially in more complex problems.

**Why verify your solutions?**

• Accuracy: Even minor calculation errors can lead to incorrect solutions, so verification acts as a check.
• Confidence: Verifying provides reassurance that the method applied and the solutions derived are correct.

To verify, simply substitute the computed values of \(u\) and \(v\) back into the original equations. Using our solved values \(u = \frac{32}{17}\) and \(v = \frac{11}{17}\):
1. For the first equation: \(2u + 5v = 7\), we find \(2\left(\frac{32}{17}\right) + 5\left(\frac{11}{17}\right) = \frac{119}{17} = 7\). 2. For the second equation: \(3u - v = 5\), substitute to get \(3\left(\frac{32}{17}\right) - \frac{11}{17} = \frac{85}{17} = 5\). Both substitutions satisfy their respective equations, confirming the solutions as correct.