Problem 24 - Intermediate Algebra

Question

For Problems \(1-26\), solve each system by using the substitution method. (Objective 1) $$ \left(\begin{array}{l} 2 x+3 y=3 \\ 4 x-9 y=-4 \end{array}\right) $$

Step-by-Step Solution

Answer

\(x = \frac{1}{2}\), \(y = \frac{2}{3}\)

Step 1: Solve for one variable

To use the substitution method, start by solving one of the equations for one of the variables. From the first equation, \(2x + 3y = 3\), solve for \(x\):\[2x = 3 - 3y\]\[x = \frac{3 - 3y}{2}\]

Step 2: Substitute into the second equation

Substitute the expression for \(x\) from Step 1 into the second equation, \(4x - 9y = -4\):\[4\left(\frac{3 - 3y}{2}\right) - 9y = -4\]

Step 3: Simplify and solve for \(y\)

Simplify the expression from Step 2:\[4\left(\frac{3 - 3y}{2}\right) = 2(3 - 3y) = 6 - 6y\]Substitute back:\[6 - 6y - 9y = -4\]\[6 - 15y = -4\]\[-15y = -10\]\[y = \frac{-10}{-15} = \frac{2}{3}\]

Step 4: Solve for \(x\) using \(y\)

Now that we have \(y = \frac{2}{3}\), substitute it back into the expression for \(x\):\[x = \frac{3 - 3\times\frac{2}{3}}{2}\]\[x = \frac{3 - 2}{2} = \frac{1}{2}\]

Step 5: Solution of the system

The solution to the system of equations is \(x = \frac{1}{2}\) and \(y = \frac{2}{3}\).

Key Concepts

System of Equations

In mathematics, a system of equations is a collection of two or more equations with the same set of unknowns. In our example, we have two equations with two variables, namely \(x\) and \(y\). This means that we need to find pairs of values for \(x\) and \(y\) that satisfy both equations simultaneously.

A system of equations can be solved through various methods, one of which is the substitution method. This particular system of equations can be written as:

• 2x + 3y = 3
• 4x - 9y = -4

Each equation within the system can be imagined as a line on a graph. The solution to the system is the point where both lines intersect. In our given exercise, the intersection point of these lines is the solution we are seeking, represented as \(x = \frac{1}{2}\) and \(y = \frac{2}{3}\).

The goal is to find a value for each variable that satisfies all the equations in the system.

Algebraic Solution

Algebraic solutions involve manipulating equations to find unknown values. In the context of a system of equations, algebraic solution refers to the process of finding the values of unknowns that satisfy all equations simultaneously.

One practical algebraic method is the substitution method, which involves solving one of the equations for one variable, and then substituting this expression into the other equations. This can help eliminate one variable, allowing us to solve for the remaining unknowns step by step.

In our problem, we start by solving the first equation for \(x\):

• 2x + 3y = 3
• \(x = \frac{3 - 3y}{2}\)

Substituting this expression for \(x\) in the second equation gives a single equation with one unknown, which is simpler to solve:

• 4\(x\) - 9y = -4
• \(4(\frac{3 - 3y}{2}) - 9y = -4\)

This substitution allows us to solve for \(y\) first and then use that value to find \(x\). This strategic elimination of variables is key to efficiently solving systems of equations.

Linear Equations

Linear equations are equations of the first order that graph as straight lines. The general form of a linear equation with two variables is \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. Our given system consists of two such linear equations.

Characteristics of linear equations useful in solving systems include:

• They have at most one solution if they intersect at a single point.
• They may have infinitely many solutions if they are identical (overlapping lines).
• They may have no solution if they are parallel.

In the substitution method, our purpose is to derive these qualities algebraically rather than graphically. By rearranging and solving these equations, we make use of the properties of linear equations—including their constant rate of change, represented by the coefficients \(a\) and \(b\).

Ultimately, solving the system using substitution leads us to intercepts at \(x = \frac{1}{2}\) and \(y = \frac{2}{3}\), confirming that these are distinct lines that intersect at a single point.