Problem 21 - Precalculus Mathematics for Calculus

Question

Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered pair form given in Example 6. $$\left\\{\begin{array}{r}x+y=4 \\\\-x+y=0\end{array}\right.$$

Step-by-Step Solution

Answer

The unique solution is \((2, 2)\).

Step 1: Understand the System of Equations

The given system of equations is: \[ \begin{cases} x + y = 4 & \text{(Equation 1)} \ -x + y = 0 & \text{(Equation 2)} \end{cases} \] Our goal is to find the values of \( x \) and \( y \) that satisfy both equations.

Step 2: Solve Equation 2 for y

Equation 2 is \(-x + y = 0\). Solve for \(y\): \[ y = x \] This expresses \(y\) in terms of \(x\).

Step 3: Substitute y in Equation 1

Substitute \(y = x\) from Step 2 into Equation 1:\[ x + (x) = 4 \]This simplifies to \[ 2x = 4 \].

Step 4: Solve for x

Solve the equation \(2x = 4\) for \(x\):\[ x = \frac{4}{2} \]\[ x = 2 \]

Step 5: Find y using x=2

Use the expression for \(y\) from Step 2, \(y = x\), and substitute the value of \(x = 2\):\[ y = 2 \]

Step 6: Verify the Solution

Verify that \(x = 2\) and \(y = 2\) satisfy both original equations:- For Equation 1: \(2 + 2 = 4\) which is true.- For Equation 2: \(-2 + 2 = 0\) which is true.Both equations are satisfied, confirming the solution is correct.

Key Concepts

Linear Equations

A linear equation is a type of equation where the variables are raised only to the power of one. It forms a straight line when graphed on a coordinate plane. In our exercise, we have two linear equations given as:

• Equation 1: \( x + y = 4 \)
• Equation 2: \( -x + y = 0 \)

Each equation represents a straight line. To solve for the variables \(x\) and \(y\), we need to find the point where these two lines intersect on the coordinate plane. This point is known as the solution to the system of equations.

Solving Algebraically

Solving algebraically refers to finding the variable values that satisfy the given equations using algebraic manipulations. For the system of equations, it's often easier to solve one of the equations for a single variable and then substitute it into the other equation.
In our example, we first solved Equation 2 for \(y\):

• \( -x + y = 0 \)
• Adding \(x\) to both sides gives: \( y = x \)

Now, we have expressed \(y\) in terms of \(x\), which helps us reduce the number of variables in the equations, making it easier to solve the entire system.

Substitution Method

The substitution method is a widely used approach to solve systems of equations. It involves isolating one variable in one of the equations and then substituting it into the other equation. This method simplifies the system to a single equation with one variable.
In the given problem, using the substitution method involved substituting \( y = x \) (from our solution of Equation 2) back into Equation 1:

• Substituting gives: \( x + (x) = 4 \)
• This equation simplifies to \( 2x = 4 \)
• Divide both sides by 2 to solve for \(x\): \( x = 2 \)

Once we found \(x\), we used the initial substitution \( y = x \) to find \( y = 2 \). The substitution method is powerful because it simplifies complex systems into smaller, more manageable problems.