Problem 4 - Algebra Form and Function

Question

Solve the systems of equations. $$ \left\\{\begin{array}{l} x=y-9 \\ 4 x-y=0 \end{array}\right. $$

Step-by-Step Solution

Answer

Answer: The solution of the given system of equations is (3, 12).

Step 1: Write the equations

The given system of equations is as follows: $$ \left\\{\begin{array}{l} x=y-9 \\\ 4 x-y=0 \end{array}\right. $$

Step 2: Substitute x with the expression from the first equation

The first equation can be rewritten as \(x = y - 9\). We'll substitute this into the second equation to eliminate x. Now the second equation becomes: $$4(y-9) - y = 0$$

Step 3: Solve the resulting equation for y

Next, we'll solve the equation from Step 2 for y. \begin{align*} 4(y-9) - y &= 0 \\ 4y - 36 - y &= 0 \\ 3y - 36 &= 0 \\ 3y &= 36 \\ y &= 12 \end{align*} Now we've found the value of y, which is 12.

Step 4: Substitute the value of y into the first equation to find x

Now that we have the value of y, we can substitute it into the first equation to find the value of x. \begin{align*} x &= y - 9 \\ x &= 12 - 9 \\ x &= 3 \end{align*} Now we've found the value of x, which is 3.

Step 5: Write the solution as an ordered pair

Finally, we can write the solution of the system as an ordered pair (x, y): $$(3, 12)$$ The solution of the given system of equations is \((3,12)\).

Key Concepts

Substitution Method

The substitution method is a powerful technique used for solving systems of equations. It involves replacing one variable with an expression involving the other variable to simplify the system.
This method is particularly useful when one of the equations is already solved for one of the variables, or can be easily manipulated to do so.

• Start by solving one of the equations for one of the variables. In our given system, this means rewriting the first equation as \(x = y - 9\).
• Take the expression \(x = y - 9\) and substitute it into another equation to eliminate the variable \(x\). This substitution simplifies the system to only one variable.
• After the substitution, solve the resulting single-variable equation to find the value of the remaining variable.

This step-by-step substitution not only aids in deciphering complex equations but also ensures accuracy in solutions.

Solving Linear Equations

Solving linear equations is about finding the value of the variables that make the equation true. Linear equations typically take the form \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants.
For example, in this system, after substitution, the equation becomes \(4(y - 9) - y = 0\).

• First, simplify the equation using basic algebraic operations like distribution, addition, or subtraction. This might involve combining like terms to make the equation easier to manage.
• Isolate the variable on one side of the equation. For instance, combining like terms in \(4y - 36 - y = 0\) simplifies to \(3y - 36 = 0\).
• Finally, solve for the variable to find its value. For our example, isolate \(y\) to find \(y = 12\).

Understanding linear equations is fundamental because they form the foundation for tackling more complex algebraic challenges.

Simultaneous Equations

Simultaneous equations are a set of equations with multiple unknowns that satisfy all equations at the same time. They are commonly encountered in algebra, and solving them identifies the point where each equation intersects.
Our goal is to find values of \(x\) and \(y\) that satisfy both equations.

• In the given system, after finding \(y = 12\), substitute back into one of the original equations, \(x = y - 9\), to find \(x\). This yields \(x = 3\).
• The solution \((3, 12)\) is a pair of values for both variables that make each equation true simultaneously.
• This approach, essentially, finds where both equations intersect graphically.

Solving simultaneous equations not only strengthens mathematical intuition but also aids in solving practical problems where multiple conditions must be met at once.