Problem 35 - Intermediate Algebra

Question

Solve each system by elimination. See Example 5 . $$ \left\\{\begin{array}{l} 2 x-\frac{5}{2}=y \\ 0.04 x-0.02 y=0.05 \end{array}\right. $$

Step-by-Step Solution

Answer

The system has infinitely many solutions with \( y = 2x - \frac{5}{2} \).

Step 1: Express y in terms of x

The first equation is given as \( y = 2x - \frac{5}{2} \). This expression for \( y \) will be used to eliminate \( y \) from the second equation.

Step 2: Substitute y into the second equation

Substitute \( y = 2x - \frac{5}{2} \) into the second equation \( 0.04x - 0.02y = 0.05 \). This gives us: \( 0.04x - 0.02(2x - \frac{5}{2}) = 0.05 \).

Step 3: Simplify the equation

Distribute \(-0.02\) in the equation: \( 0.04x - 0.04x + 0.02 \times \frac{5}{2} = 0.05 \). This simplifies to \( 0.05 = 0.05 \). The equation holds true in all scenarios, indicating infinitely many solutions.

Step 4: Check for consistent solutions

Since the second equation reduced to a true statement, it implies that the two original equations are dependent on each other. Thus, the solution set of the system is determined by any \( x \) and \( y = 2x - \frac{5}{2} \).

Key Concepts

Elimination Method

The elimination method is a popular strategy for solving systems of equations. This method involves "eliminating" one of the variables by aligning two equations in such a way that addition or subtraction of the equations cancels out that variable. To do this, you may need to multiply one or both of the equations by a constant to ensure that the coefficients of one of the variables are opposites. When the variable is eliminated, you are left with an equation in one variable, which is much simpler to solve. This method is particularly useful when the coefficients or numbers in the equations lend themselves easily to cancellation through addition or subtraction.

Dependent Equations

Dependent equations are equations that rely on each other for their solutions. This means that both equations represent the same line or have all points in common. In such cases, one equation is essentially a rearranged or scaled version of the other. When solving a system of equations involving dependent equations, you are not trying to find a single solution pair. Instead, you identify that the two equations are representations of the same line. In a graph, dependent equations would appear as a single line, not two distinct lines.

Infinitely Many Solutions

A system of equations with infinitely many solutions means that there are endless combinations of variables that satisfy both equations simultaneously. In the case of linear equations, this typically occurs with dependent equations where every point on the line is a solution. Mathematically, when solving, you might simplify the equations to an identity, such as "0 = 0," confirming that any value for one variable can correspond to many values of the other variable. This is a key indicator that the system is not just consistent but also dependent. Understanding this concept helps clarify why some systems do not yield a single solution but rather an entire range of possibilities.

Substitution Method

The substitution method is another effective way to solve systems of equations. It involves isolated one variable in one equation, and then substituting that expression into the other equation. By doing so, you convert a system of two equations into a single equation with one variable, making the problem more manageable. After solving this single-variable equation, you substitute the obtained value back into any of the original equations to find the corresponding variable. This method is particularly useful when one equation is already solved for a single variable, making it easier to substitute and solve.