Problem 60 - Algebra for College Students

Question

Solve each system by using either the substitution method or the elimination- by-addition method, whichever seems more appropriate. \(\left(\begin{array}{l}2 x+y=7.75 \\ 3 x+2 y=12.5\end{array}\right)\)

Step-by-Step Solution

Answer

The solution is \((x, y) = (3, 1.75)\).

Step 1: Choose the Method

For this system of equations, it is preferable to use the substitution method because we can easily solve for one variable and substitute it into the other equation. Let's solve for \( y \) from the first equation: \( 2x + y = 7.75 \).

Step 2: Solve for one Variable

From the first equation, \( 2x + y = 7.75 \), solve for \( y \):\[ y = 7.75 - 2x \]

Step 3: Substitute into the Other Equation

Now, substitute \( y = 7.75 - 2x \) into the second equation \( 3x + 2y = 12.5 \):\[ 3x + 2(7.75 - 2x) = 12.5 \]

Step 4: Simplify the Equation

Simplify and solve the equation:\[ 3x + 15.5 - 4x = 12.5 \] \[ -x + 15.5 = 12.5 \] \[ -x = -3 \]

Step 5: Solve for x

Solve for \( x \) by dividing both sides by -1:\[ x = 3 \]

Step 6: Substitute back to find y

Use the value of \( x \) found to calculate \( y \):Plug \( x = 3 \) into \( y = 7.75 - 2x \):\[ y = 7.75 - 2(3) \] \[ y = 7.75 - 6 \] \[ y = 1.75 \]

Step 7: Check the Solution in Both Equations

Verify that the solution \( (x, y) = (3, 1.75) \) satisfies both original equations.Check: \( 2x + y = 7.75 \) becomes \( 2(3) + 1.75 = 6 + 1.75 = 7.75 \), and\( 3x + 2y = 12.5 \) becomes \( 3(3) + 2(1.75) = 9 + 3.5 = 12.5 \).Both equations are satisfied.

Key Concepts

Substitution Method

The substitution method is a fantastic approach to solving a system of linear equations. It's particularly useful when one of the equations can be easily solved for one variable. Here's how it works: you solve one of the equations for one variable, and then substitute this expression into the other equation. This way, you end up with an equation that contains only one variable.

• First, choose which equation to manipulate. Usually, it's the one where you can easily solve for a variable.
• Solve the chosen equation for one of the variables.
• Substitute this solution into the other equation. This will give you an equation with just one unknown to solve.

For instance, in our original exercise, we solved for \( y \) in the first equation \( 2x + y = 7.75 \), resulting in \( y = 7.75 - 2x \). We then substituted this into the second equation, \( 3x + 2y = 12.5 \). This substitution transforms the problem into solving a single-variable linear equation.

Elimination Method

The elimination method is another strategy for solving systems of linear equations. In this method, the goal is to eliminate one variable by adding or subtracting the equations. Adjust the coefficients if needed so that when you add or subtract the equations, one variable cancels out. This method is particularly useful if both equations have similar coefficients.

• Align equations such that the variables and their coefficients are in columns.
• Multiply one or both of the equations by a number to get the coefficients of one variable to be opposite numbers.
• Add or subtract the equations to eliminate one variable.

By eliminating one variable, you simplify the system to a single equation with one unknown. This was not used in our problem since the substitution method was more straightforward in this case, but it's good to know multiple approaches!

Solving Linear Equations

Once the system is reduced to a single linear equation, whether by substitution or elimination, the next step is solving it for the unknown variable. This involves manipulating the equation to isolate the variable on one side of the equation.

• Start by simplifying: combine like terms and simplify any expressions.
• Isolate the variable: use addition, subtraction, multiplication, or division to get the variable alone on one side of the equation.

In our exercise, after substituting the expression for \( y \), we had \( 3x + 15.5 - 4x = 12.5 \), which simplified to \( -x = -3 \). Solving this gives \( x = 3 \). This straightforward process is crucial for finding the value of one variable before moving on to the next.

Checking Solutions

After finding values for the variables, it's essential to verify that these solutions satisfy the original equations. This step ensures accuracy and confidence in your work.

• Substitute the found values back into the original equations.
• Verify that both equations hold true with these values.

In this exercise, once we found \( x = 3 \) and \( y = 1.75 \), we substituted these into both original equations. For \( 2x + y = 7.75 \), checking gives \( 2(3) + 1.75 = 7.75 \); correct! For \( 3x + 2y = 12.5 \), it gives \( 9 + 3.5 = 12.5 \); also correct! This confirmation builds confidence that the solution is indeed correct.