Problem 20 - A Graphical Approach to College Algebra

Question

Solve each system by substitution. $$\begin{aligned}&7 x-y=-10\\\&3 y-x=10\end{aligned}$$

Step-by-Step Solution

Answer

The solution is \(x = -1\) and \(y = 3\).

Step 1: Solve One Equation for a Variable

Choose one of the equations to solve for one of the variables. Let's solve the first equation for \(y\). \[7x - y = -10\] Rearrange it to express \(y\) in terms of \(x\): \[y = 7x + 10\]

Step 2: Substitute into the Other Equation

Substitute the expression for \(y\) from Step 1 into the second equation. Start with: \[3y - x = 10\] Replace \(y\) with \(7x + 10\): \[3(7x + 10) - x = 10\]

Step 3: Simplify and Solve for x

Simplify the equation from Step 2: \[21x + 30 - x = 10\] Combine like terms: \[20x + 30 = 10\] Subtract 30 from both sides to isolate terms with \(x\): \[20x = -20\] Now, divide by 20: \[x = -1\]

Step 4: Substitute x Back to Find y

Use the value of \(x\) found in Step 3 and substitute it back into the equation derived in Step 1: \[y = 7(-1) + 10\] Simplify to find \(y\): \[y = -7 + 10 = 3\]

Step 5: Verify the Solution

Plug \(x = -1\) and \(y = 3\) back into both original equations to ensure they satisfy both: - For \(7x - y = -10\): \[7(-1) - 3 = -10\] which holds true.- For \(3y - x = 10\): \[3(3) + 1 = 10\] which also holds true.Thus, the solution is verified.

Key Concepts

Understanding a System of Equations

A system of equations is a collection of two or more equations with the same set of unknowns. In this particular exercise, we are dealing with a system of linear equations, which means the equations involve only first-degree terms, such as \( x \) and \( y \). Our goal is often to find the values of these variables that satisfy all given equations simultaneously.

In our example, the system we are working with consists of two equations:

• \( 7x - y = -10 \)
• \( 3y - x = 10 \)

These equations must both be true at the same time for the same values of \( x \) and \( y \). The point where the solutions intersect on a graph, if plotted, would represent the solution to the system.

Typically, systems can be solved using various methods such as substitution, elimination, or graphing. Here, we employ the substitution method, making one of the equations simpler to handle by solving it for one variable and substituting its value into the other equation.

Solving Linear Equations with Substitution

Solving linear equations is a fundamental skill in algebra that involves finding the value of variables that make the equation true. The substitution method is a particularly useful technique when solving systems of equations.

This method begins by solving one of the equations for one of the variables. In our exercise, we initially solve the first equation \( 7x - y = -10 \) for \( y \) :

• Rearrange to find \( y = 7x + 10 \).

Then, this expression for \( y \) is substituted into the other equation, \( 3y - x = 10 \), yielding:

• \( 3(7x + 10) - x = 10 \).

This creates an equation with a single variable, \( x \), which can then be solved more straightforwardly. Once \( x \) is determined, this value is used to find the value of \( y \), completing the solution to the system.

Substitution simplifies solving systems because it reduces the problem to solving a single linear equation at a time.

Algebraic Manipulation in Solving Equations

Algebraic manipulation is essential for solving equations and systems. It involves rearranging and simplifying expressions to isolate variables and find their values.

In our system, we used algebraic manipulation to solve one equation for a variable, substituting it into the other equation. Details of this process include:

• Expanding terms: \( 3(7x + 10) = 21x + 30 \)
• Combining like terms: \( 21x - x = 20x \)
• Isolating variables: \( 20x + 30 = 10 \) becomes \( 20x = -20 \)
• Division to solve for \( x \): \( x = -1 \)

After finding \( x \), we substitute it back into the rearranged first equation to find \( y \), using simple arithmetic:

• Substitute \( x = -1 \): \( y = 7(-1) + 10 = 3 \)

This systematic manipulation helps in clearly understanding and solving equations step-by-step. Recognizing when and how to perform these manipulations is key to mastering algebra and tackling more complex mathematical problems.