Problem 58 - Algebra 2

Question

Solve each system by elimination or substitution. $$ \left\\{\begin{array}{r}{3 x+y=4} \\ {2 x-4 y=7}\end{array}\right. $$

Step-by-Step Solution

Answer

\((x, y) = \left(\frac{23}{14}, -\frac{33}{14}\right)\)

Step 1: Choose a method to solve the system

Decide whether to use elimination or substitution to solve the system of equations. In this case, substitution might be more straightforward since the first equation can easily be solved for y.

Step 2: Solve for y in the first equation

Isolate y in the first equation: \[ y = 4 - 3x \]

Step 3: Substitute y into the second equation

Substitute 4 - 3x in for y in the second equation: \[ 2x - 4(4 - 3x) = 7 \]

Step 4: Solve for x

Distribute and combine like terms then solve for x: \[ 2x - 16 + 12x = 7 \] \[ 14x = 23 \] \[ x = \frac{23}{14} \]

Step 5: Substitute x back into the first equation to find y

Substitute \( x = \frac{23}{14} \) into the first equation and solve for y: \[ 3\left(\frac{23}{14}\right) + y = 4 \] \[ y = 4 - \frac{69}{14} \] \[ y = -\frac{33}{14} \]

Step 6: Write the solution as an ordered pair

Combine the values of x and y to present the solution as an ordered pair, which represents the point of intersection of the two lines: \[ (x, y) = \left(\frac{23}{14}, -\frac{33}{14}\right) \]

Key Concepts

Understanding the Elimination Method

Solving systems of equations using the elimination method involves manipulating the equations so that one of the variables can be eliminated when the equations are combined. This method is particularly useful when both equations in the system can be easily manipulated to have one of the variables cancel when added or subtracted.

To apply the elimination method, you can multiply the equations by appropriate numbers so that either the coefficients of the x's or the y's are opposites. Let's consider a situation where we multiply the first equation, 3x + y = 4, by 4 and the second equation, 2x - 4y = 7, by 1, which gives us new equations with the same 'y' coefficient, but opposite signs. When we add these two equations, the 'y' terms cancel out:

• 12x + 4y = 16
• 2x - 4y = 7

Adding the two results in:
14x + 0y = 23
Now, we can solve for x as we have a single variable equation.

Note that the elimination method might require you to multiply the equations by different numbers if they are not set up for easy elimination. Always aim to obtain coefficients that will cancel one variable when you combine the equations.

The Substitution Method Explained

On the other hand, the substitution method entails rearranging one of the equations to solve for one variable in terms of the other, and then substituting this value into the second equation. This turns a system of two equations into just one equation with one variable.

For instance, in our given system, the first equation, 3x + y = 4, can be rearranged to solve for y, giving us y = 4 - 3x. This expression for y is then substituted into the other equation, replacing every 'y' with 4 - 3x. The resulting equation, 2x - 4(4 - 3x) = 7, is solved like any other single-variable equation to find the value of x.

After finding the value of x, this number is then substituted back into either original equation to find the corresponding value of y. Hence, both the x and y values are found one after the other, allowing us to state the solution as an ordered pair that represents the intersection of the two lines described by the original equations.

Ordered Pair Solution Significance

The final step in solving systems of equations is writing the solution as an ordered pair, \(x, y\). This ordered pair represents the exact point at which the two lines intersect on a graph. When obtaining the solution as an ordered pair, it's essential to ensure the values satisfy both original equations.

In our example, the solution to the system is \(x, y\) = \left(\frac{23}{14}, -\frac{33}{14}\right)\. To verify that this is the correct solution, you can substitute these values of x and y back into the original equations to see if both equations are true with these numbers. If they are, you have found the point of intersection, and your solution is correct.

An ordered pair is not just an answer; it is a key concept in algebra that links algebraic solutions to geometric interpretations. Thus, any problem involving systems of equations is not just about finding x and y separately; it's about understanding their interaction and the point they describe collectively on the Cartesian plane.