Question

Solve each system of equations. If the system has no solution, state that it is inconsistent. $$ \left\\{\begin{array}{l} 3 x-2 y=0 \\ 5 x+10 y=4 \end{array}\right. $$

Short answer

\(x = \frac{1}{5}, y = \frac{3}{10}\)

Step-by-step solution

- Write down the system of equations

The given system of equations is: \[\begin{array}{l} 3x - 2y = 0 \ 5x + 10y = 4 \end{array}\]

- Solve the first equation for one variable

Solve the first equation for x: \[3x - 2y = 0\] Add 2y to both sides: \[3x = 2y\] Now divide by 3: \[x = \frac{2y}{3}\]

- Substitute the expression into the second equation

Substitute \( x = \frac{2y}{3} \) into the second equation: \[5\frac{2y}{3} + 10y = 4\]

- Solve the resulting equation

Multiply through by 3 to clear the fraction: \[5 * 2y + 30y = 12\] Simplify: \[10y + 30y = 12\] Combine like terms: \[40y = 12\] Divide by 40: \[y = \frac{12}{40}\] Simplify: \[y = \frac{3}{10}\]

- Find the value of x

Substitute \( y = \frac{3}{10} \) back into the expression \( x = \frac{2y}{3} \): \[x = \frac{2(\frac{3}{10})}{3}\] Simplify: \[x = \frac{6}{30}\] \[x = \frac{1}{5}\]

- State the solution

The solution to the system of equations is: \[x = \frac{1}{5}, y = \frac{3}{10}\]

Key concepts

linear equations

Linear equations are equations where the highest power of the variable is 1. These equations form straight lines when graphed on a coordinate plane. The general form of a linear equation in two variables is: \[ax + by = c\]where \(a\), \(b\), and \(c\) are constants. In the exercise given, we have two linear equations forming a system. The goal is to find values for \(x\) and \(y\) that satisfy both equations simultaneously. Linear equations can be solved using various methods such as substitution, elimination, and graphing. Here, we'll focus on the substitution method.

substitution method

The substitution method is a technique to solve systems of equations by solving one of the equations for one variable and then substituting this expression into the other equation. This method transforms a system of two equations into a single equation in one variable. Here’s a step-by-step outline of how it works:
1. Solve one of the equations for a single variable. For example, solve for \(x\) in the first equation: \[3x - 2y = 0 \rightarrow x = \frac{2y}{3}\]
2. Substitute the solved expression into the other equation. In this case, substitute \(x = \frac{2y}{3}\) into the second equation: \[5(\frac{2y}{3}) + 10y = 4\]
3. Simplify and solve for the remaining variable: \[\frac{10y}{3} + 30y = 12\rightarrow 40y = 12\rightarrow y = \frac{3}{10}\]
4. Substitute back the value of the solved variable into one of the original equations to find the other variable: \[x = \frac{2(\frac{3}{10})}{3}\rightarrow x = \frac{1}{5}\]
The substitution method is particularly useful when one equation is easily solved for one variable and then substituted into the other.

inconsistent systems

An inconsistent system of equations is one that has no solution. This usually happens when the equations represent parallel lines, which never intersect. If after substituting and simplifying you end up with a false statement (e.g., \(0 = 5\)), this indicates that the system is inconsistent. For the example given: \[3x - 2y = 0\] and \[5x + 10y = 4\],when solved, we find that \(x = \frac{1}{5}\) and \(y = \frac{3}{10}\), hence the system is consistent because we have found a solution. When graphing, consistent systems will intersect at a single point or be the same line (infinitely many solutions). In contrast, inconsistent systems will show parallel lines with no points of intersection, indicating no common solution exists.