Problem 26 - 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. $$\left\\{\begin{array}{rr} x+y= & 7 \\ 2 x-3 y= & -1 \end{array}\right.$$

Step-by-Step Solution

Answer

The system has a unique solution: (4, 3).

Step 1: Write the System of Equations

We have the system of linear equations: \( \begin{cases} x+y = 7 \ 2x-3y = -1 \end{cases} \). Our goal is to solve for \(x\) and \(y\).

Step 2: Express y in terms of x from the first equation

From the first equation, solve for \(y\): \(x + y = 7 \), which gives us \(y = 7 - x\).

Step 3: Substitute y in the second equation

Substitute \(y = 7 - x\) into the second equation, \(2x - 3y = -1\): \(2x - 3(7-x) = -1\).

Step 4: Simplify and solve for x

Distribute and simplify the equation \(2x - 21 + 3x = -1\). This simplifies to \(5x - 21 = -1\), and adding 21 to both sides gives \(5x = 20\). Divide by 5 to solve for \(x\): \(x = 4\).

Step 5: Find the value of y

Use \(x = 4\) in the equation \(y = 7 - x\): \(y = 7 - 4 = 3\).

Step 6: Write the solution as an ordered pair

The solution to the system of equations is the ordered pair \((x, y) = (4, 3)\).

Step 7: Verify the solution

Check the solution by substituting \(x = 4\) and \(y = 3\) back into the original equations: \(4 + 3 = 7\) (True) and \(2(4) - 3(3) = 8 - 9 = -1\) (True). Both equations are satisfied.

Key Concepts

Linear Equations

Linear equations are one of the simplest forms of equations in algebra. They represent straight lines when graphed on a coordinate plane. In a linear equation, each term is the product of a constant and a single variable. These equations do not include variables raised to any power higher than one.
This is important because it signifies that solving linear problems involves relatively straightforward arithmetic.
A typical linear equation can be written in the standard form:

• For one variable: \( ax + b = 0 \)
• For two variables: \( ax + by = c \)

Our original exercise consists of two linear equations. Each equation involves a combination of variables \(x\) and \(y\). Solving them requires finding a common point where both lines intersect. This intersection corresponds to the solution of the system.

Substitution Method

The substitution method is a common algorithm for solving systems of equations, particularly well-suited for linear equations. The core idea is to express one variable in terms of another and substitute it back into the other equation. This reduces the problem into solving a single equation.
In our exercise, we first rearrange the equation \(x + y = 7\) to express \(y\) in terms of \(x\):

• \(y = 7 - x\)

The next step is to substitute \(y = 7 - x\) into the second equation \(2x - 3y = -1\). This substitution allows us to eliminate \(y\) from the equation system, leading to one equation with only \(x\). Solving that equation gives us \(x = 4\), which we then use to find \(y = 3\).
The substitution method is particularly useful:

• When the system involves simple linear equations.
• If one equation can be easily rearranged to express a single variable.

Ordered Pairs

In mathematics, ordered pairs are used to convey the specific location of a point on a plane or in space, typically represented as \((x, y)\) for two-dimensional problems. Each number in the ordered pair corresponds to one of the coordinates. The order is crucial, as it defines which number goes along each axis on the coordinate plane.
Ordered pairs are a standard way to write solutions for a system of two linear equations, reflecting the values of the variables where the two equations intersect.
For the given exercise, the ordered pair solution is \((4, 3)\). This means that substituting \(x = 4\) and \(y = 3\) back into both original equations satisfies them both. The solution confirms that these values lie on both lines represented by the equations.

• Verification involves ensuring both equations hold true with these values.
• In this case, \(4 + 3 = 7\) and \(2(4) - 3(3) = -1\) both equal the respective right side of the equations.