Problem 7 - Algebra and Trigonometry

Question

\(5-8\) Use the substitution method to find all solutions of the system of equations. $$ \left\\{\begin{array}{c}{x-y=2} \\ {2 x+3 y=9}\end{array}\right. $$

Step-by-Step Solution

Answer

The solution is \((x, y) = (3, 1)\).

Step 1: Solve the First Equation for x

Start by solving the first equation for \( x \). The first equation is \( x - y = 2 \). Add \( y \) to both sides to isolate \( x \):\[ x = y + 2 \]

Step 2: Substitute x in the Second Equation

Now that we have \( x = y + 2 \), substitute \( x \) in the second equation \( 2x + 3y = 9 \) with \( y + 2 \):\[ 2(y + 2) + 3y = 9 \]

Step 3: Simplify and Solve for y

Distribute the \( 2 \) in the equation \( 2(y + 2) \):\[ 2y + 4 + 3y = 9 \]Combine like terms (\( 2y \) and \( 3y \)):\[ 5y + 4 = 9 \]Subtract \( 4 \) from both sides to isolate the term with \( y \):\[ 5y = 5 \]Divide both sides by \( 5 \) to solve for \( y \):\[ y = 1 \]

Step 4: Substitute y back into the x equation

Now substitute \( y = 1 \) back into the equation \( x = y + 2 \):\[ x = 1 + 2 \]Simplify:\[ x = 3 \]

Step 5: Write the Solution

The solution to the system of equations is the pair \( (x, y) \). After solving, we found: \( x = 3 \) and \( y = 1 \). Thus, the solution is:\[ (x, y) = (3, 1) \]

Key Concepts

System of Equations

A system of equations is a collection of two or more equations with the same set of unknowns. In mathematics, these equations are dealt with simultaneously to find a common solution. In this particular exercise, we have a system composed of two linear equations:

• The first equation: \( x - y = 2 \)
• The second equation: \( 2x + 3y = 9 \)

By solving these equations together, you deduce values for \( x \) and \( y \) that make both equations true simultaneously. The solution to a system of linear equations can be found using various methods, such as substitution, elimination, or graphical representation. Here, the substitution method is applied, allowing us to solve one equation for a variable, and then replacing that variable in the other equation.

Linear Equations

Linear equations are algebraic expressions that describe a straight line when plotted on a graph. They have variables raised only to the first power and can be written in the form of \( ax + by = c \). Here, 'a', 'b', and 'c' are constants.In our example:

• \( x - y = 2 \) is a linear equation representing a line with a slope of 1 and a y-intercept at \( -2 \).
• \( 2x + 3y = 9 \) depicts another line with a different slope, involved in our system of equations.

These lines intersect at a particular point, which represents the solution of the system. Linear equations form the basis of solving systems because their solutions are straightforward and predictable.

Solution of Equations

The solution of a system of equations refers to the values for the variables that satisfy all equations in the system simultaneously. Using the substitution method, we first solved one equation for \( x \), finding \( x = y + 2 \). This expression for \( x \) was then substituted into the second equation.After substitution and simplification:
- We arrived at \( 5y + 4 = 9 \)- Solving for \( y \), we found \( y = 1 \)- Substituting \( y = 1 \) back into the equation for \( x \), we derived \( x = 3 \)The intersection point \((3, 1)\) is the solution, showing where the lines representing the equations cross each other. This coordinate pair is the set of values that solve both equations at the same time, confirming the accuracy of the solution obtained using the substitution method.