Problem 13 - Precalculus: Mathematical 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 3. $$\left\\{\begin{array}{rr} -x+y= & 2 \\ 4 x-3 y= & -3 \end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system is \((3, 5)\).

Step 1: Convert system into equations

Let's write the given system of equations as two separate equations. The first equation is \(-x + y = 2\). The second equation is \(4x - 3y = -3\).

Step 2: Solve first equation for y

From the first equation \(-x + y = 2\), we can solve for \(y\) by adding \(x\) to both sides:\[y = x + 2\].

Step 3: Substitute y into the second equation

Using the expression for \(y\) from Step 2, substitute \(y = x + 2\) into the second equation \(4x - 3y = -3\):\[4x - 3(x + 2) = -3\].

Step 4: Simplify and solve for x

Expand and simplify the equation from Step 3:\[4x - 3x - 6 = -3\]Combine like terms:\[x - 6 = -3\]Add 6 to both sides:\[x = 3\].

Step 5: Substitute x back to find y

Now, substitute \(x = 3\) back into the equation found in Step 2 \(y = x + 2\):\[y = 3 + 2 = 5\].

Step 6: Write the solution as an ordered pair

The solution to the system of equations is the ordered pair \((x, y) = (3, 5)\). This means the point \((3, 5)\) satisfies both original equations.

Key Concepts

System of Equations

A system of equations consists of two or more equations that share the same set of variables. In the context of linear equations, these equations form lines on a graph, and the solution to the system corresponds to the point where these lines intersect. For example, consider the original system:

• \(-x + y = 2\)
• \(4x - 3y = -3\)

These equations must be satisfied simultaneously for the solution to be valid. The intersection point represents the values of \(x\) and \(y\) that satisfy both equations. If the lines intersect at a single point, the system has a unique solution. If the lines are parallel, there is no solution since they never meet. If they are coincident (overlapping entirely), there are infinitely many solutions since every point on the line fits both equations.

Substitution Method

The Substitution Method is a common strategy for solving systems of equations. It involves solving one of the equations for one variable and then substituting that expression into the other equation. Here’s a step-by-step explanation:

1. **Solve one of the equations for a variable:** In our example, we solved the first equation \(-x + y = 2\) for \(y\), yielding \(y = x + 2\).

2. **Substitute the expression into the other equation:** Replace \(y\) in the second equation \(4x - 3y = -3\) with \(x + 2\). This results in \(4x - 3(x + 2) = -3\).

3. **Simplify and solve for the remaining variable:** Simplifying gives \(x = 3\), which solves one of the variables.

4. **Plug back to find the other variable:** Substituting \(x = 3\) back into \(y = x + 2\) gives \(y = 5\).

The substitution method is particularly useful when one equation is simple to manipulate. It helps in reducing the system to one equation with one variable, making it easier to solve. This method is efficient and minimizes personal errors during calculations.

Ordered Pair Solution

An ordered pair solution is a pair of numbers that defines the values of variables \((x, y)\) that satisfy both equations in a system. Once the values are found, they are expressed as \((3, 5)\), indicating that when \(x = 3\) and \(y = 5\), both equations are true. This solution can be verified by substituting these values back into the original equations.

Consider our example:

• Substitute \((x, y) = (3, 5)\) into \(-x + y = 2\), which simplifies to \(-3 + 5 = 2\), proving true.
• Substitute into the second equation, \(4x - 3y = -3\), simplifying to \(12 - 15 = -3\), again proving true.

Thus, the ordered pair \((3, 5)\) is the point of intersection of the two lines represented by the equations and confirms that it is the correct solution to the system. Representing solutions as ordered pairs allows them to be easily communicated and understood geometrically as points on a graph.