Problem 13 - Algebra and Trigonometry

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 3. $$\left\\{\begin{array}{l}{-x+y=2} \\ 4 x-3 y=-3\end{array}\right.$$

Step-by-Step Solution

Answer

The system has a unique solution: \((3, 5)\).

Step 1: Analyze the Equations

The given system of equations is: \(-x + y = 2\) and \(4x - 3y = -3\). We need to find values of \(x\) and \(y\) that satisfy both equations or determine if there is no solution or infinitely many solutions.

Step 2: Determine a Suitable Method

We will use the substitution method. First, we solve the first equation for \(y\) in terms of \(x\) to substitute into the second equation.

Step 3: Solve for y in the First Equation

Rewrite the first equation \(-x + y = 2\) as \(y = x + 2\). Now, \(y\) is expressed in terms of \(x\).

Step 4: Substitute y in the Second Equation

Substitute \(y = x + 2\) into the second equation \(4x - 3y = -3\). This gives us: \[4x - 3(x + 2) = -3\].

Step 5: Simplify and Solve for x

Expand and simplify the equation: \[4x - 3x - 6 = -3\], which becomes \[x = 3\]. So, \(x = 3\).

Step 6: Find y Using the Expression

Substitute \(x = 3\) back into \(y = x + 2\). Thus, \(y = 3 + 2 = 5\).

Step 7: Confirm the Solution

Check the solution \((3, 5)\) in both original equations. First equation: \(-3 + 5 = 2\), which is valid. Second equation: \(4(3) - 3(5) = 12 - 15 = -3\), which is also valid. Thus \((3, 5)\) is indeed the solution.

Key Concepts

Substitution Method

The substitution method is a powerful technique for solving systems of linear equations. It involves solving one of the equations for one variable and then substituting that expression into the other equation. This effectively reduces the problem to a single equation in one variable, which can be solved more easily.

• First, identify which variable would be easiest to solve for in one of the equations. This could be determined by considering coefficients or the simplicity of the expression.
• Next, solve the chosen equation for that variable. For instance, if you solve for \( y \) in terms of \( x \), you may have \( y = x + 2 \).
• The final step is to substitute this expression into the other equation. This allows you to solve for the remaining variable.

After finding one variable, substitute its value back into the original expression to find the other variable. This helps ensure consistency across the solution. This method is most effective with systems where one equation is easily manipulated.

Linear Equations

Linear equations are mathematical statements that describe a linear relationship between two variables. These equations are written in the form \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) and \( y \) are variables.

• They graph to a straight line on the Cartesian plane, where each point on the line is a solution of the equation.
• Systems of linear equations can have exactly one solution, infinitely many solutions, or no solution at all. This is determined by the lines' intersection behavior in the plane.

Solving these equations involves finding the values for their variables that work for both (or all) given equations. In our context, finding the intersection point of two lines described by the equations verifies if a unique solution exists.

Ordered Pairs

Ordered pairs are a way to express solutions for the systems of equations as coordinates on the Cartesian coordinate plane. They are written in the form \((x, y)\), where \( x \) and \( y \) are specific values that satisfy the equation.

• An ordered pair represents a single point on a graph.
• For a system of equations, the ordered pair that serves as a solution is where the graphs of the equations intersect.

In our problem, after solving the system, we determined that the ordered pair \((3, 5)\) is the point where both lines intersect. This specific point satisfies both equations, confirming it as the solution.