Problem 18 - A Graphical Approach to Precalculus with Limits

Question

Solve each system by substitution. $$\begin{aligned}&3 y=5 x+6\\\&x+y=2\end{aligned}$$

Step-by-Step Solution

Answer

The solution is \((x, y) = (0, 2)\).

Step 1: Express one variable in terms of the other

We start with the equation \(x + y = 2\). Solve for \(y\) in terms of \(x\):\[ y = 2 - x \]

Step 2: Substitute expression into the other equation

Take the expression for \(y\) from Step 1 and substitute it into the first equation, \(3y = 5x + 6\):\[ 3(2 - x) = 5x + 6 \]

Step 3: Solve the equation

Distribute and simplify the equation:\[ 6 - 3x = 5x + 6 \]Move \(-3x\) to the right side:\[ 6 = 8x + 6 \]Subtract 6 from both sides:\[ 0 = 8x \]Divide by 8 to solve for \(x\):\[ x = 0 \]

Step 4: Find the value of the other variable

Use the value of \(x\) from Step 3 in the expression from Step 1:\[ y = 2 - x = 2 - 0 = 2 \]

Step 5: Verify the solution

Substitute \(x = 0\) and \(y = 2\) back into the original equations to ensure correctness. For \(3y = 5x + 6\):\[ 3(2) = 5(0) + 6 \]\[ 6 = 6 \] (True)For \(x + y = 2\):\[ 0 + 2 = 2 \] (True)

Key Concepts

System of Equations

A system of equations consists of two or more equations with the same set of variables. In this exercise, we are given two equations:

• Equation 1: \(3y = 5x + 6\)
• Equation 2: \(x + y = 2\)

The goal is to find values for \(x\) and \(y\) that satisfy both equations simultaneously. There are various methods to solve such systems, including graphing, substitution, and elimination. Each method aims to determine the common solution where the equations meet. Here, we utilize the substitution method, which involves expressing one variable in terms of the other from one of the equations and substituting it into the second equation. This method is particularly effective when dealing with linear equations.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging equations using algebraic properties to simplify them or isolate variables. In the substitution method, algebraic manipulation is essential. Let’s look at how we used it in this exercise:

• **Rearranging Equations**: From Equation 2, \(x + y = 2\), we rearranged it to express \(y\) in terms of \(x\): \(y = 2 - x\).
• **Substituting Values**: We substituted the expression for \(y\) into Equation 1, \(3(2 - x) = 5x + 6\), which involved distributing and combining like terms.

These actions allow us to reduce the system to a single variable, making it easier to solve. It is important to follow the principles of algebra cautiously to avoid errors and ensure the equations remain equivalent.

Linear Equations

Linear equations are equations of the first degree, meaning their variables are not raised to any power greater than one. Each linear equation represents a straight line when graphed on a coordinate plane. For example, \(3y = 5x + 6\) can be rewritten as \(y = \frac{5}{3}x + 2\), depicting a line with a slope of \(\frac{5}{3}\) and a y-intercept of 2.

• **Characteristics of Linear Equations**:
  • The graph of linear equations is always a straight line.
  • They have a constant rate of change, described by the slope.
  • There’s a unique intersection point in a consistent system, as seen with our solution \((0, 2)\).
  Since both equations in a system are linear, they may intersect at a single point, be parallel (no solutions), or coincide (infinitely many solutions). Substitution helps directly find the intersection point, if it exists.

Verification of Solutions

Verification is a crucial step in solving systems of equations to ensure the obtained solution is correct. After solving the equations using substitution, we obtain values for \(x\) and \(y\).

• **Substitute Back**: Replace the variables in the original equations to check if they satisfy both:
  • Substituting into \(3y = 5x + 6\): \(3(2) = 5(0) + 6\), which simplifies to \(6 = 6\), a true statement.
  • Substituting into \(x + y = 2\): \(0 + 2 = 2\), also a true statement.

Confirming your solution with the original equations validates that you have found the correct intersection of the lines. Always remember to verify solutions, as it provides confidence in your results and identifies any computational errors.