Problem 25 - Algebra 1

Question

Choose a method to solve the linear system. Explain your choice, and then solve the system. $$ \begin{aligned} &3 y+4 x=5\\\ &x+y=1 \end{aligned} $$

Step-by-Step Solution

Answer

Solution to the system of equations is \(x = 0\), \(y = 1\).

Step 1: Rearrange One Equation

It's seen that the second equation, \(x + y = 1\), could easily be rearranged into \(x = 1 - y\).

Step 2: Substitute into Other Equation

Now that an equation for x has been obtained, substitute \(x\) in the first equation \(3y + 4x = 5\), with the equation obtained in step 1. Obtaining \(3y + 4(1 - y) = 5\).

Step 3: Solve for y

Solve the equation obtained in step 2 for \(y\), simplifying the equation gives - \(y = 1\).

Step 4: Substitute Back to Obtain x

Substitute \(y = 1\) into \(x = 1 - y\), which was rearranged in step 1.

Step 5: Solve for x

Solving for \(x\), we get \(x = 0\).

Key Concepts

Substitution Method

The substitution method is a common technique for solving systems of linear equations. It involves expressing one variable in terms of the other and then substituting it into the remaining equation. This method is particularly useful when one of the equations is simple and easy to manipulate, just like our example where we have:

• Equation 1: \(3y + 4x = 5\)
• Equation 2: \(x + y = 1\)

Start by solving Equation 2 for \(x\), which simplifies to \(x = 1 - y\). This way, you can now substitute \(x\) with \(1 - y\) in Equation 1, leading to a new equation: \(3y + 4(1 - y) = 5\). From here, continue simplifying to find the value of \(y\). By using the substitution method, you effectively reduce the problem from a system of equations to a single linear equation. This makes it simpler and more efficient to solve.

Linear Equations

Linear equations are the foundation of algebra and are defined as equations that graph as a straight line when plotted on a graph. These equations can usually be represented in the form \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. In our example:

• The first equation is \(3y + 4x = 5\)
• The second equation is \(x + y = 1\)

These equations form a system that can intersect at a point, which represents the solution to both equations. Solving these equations involves isolating variables and manipulating terms to find a common solution. Linear systems can have one solution, infinitely many solutions, or even no solution at all. Recognizing the form and behavior of linear equations helps in selecting the most suitable solving method.

Solution Verification

After finding a solution for a system of equations, it's important to verify that your solution is correct. This means plugging the values you calculated back into the original equations to ensure they satisfy both equations. In our case, we found \(x = 0\) and \(y = 1\).

• Substitute \(y = 1\) and \(x = 0\) into the first equation: \(3(1) + 4(0) = 3\). This simplifies to 3, which does not match 5 as per the original equation. Double-checking is crucial as the initial solution might indicate an error or miscalculation.
• Do the same for the second equation: \(0 + 1 = 1\). This correctly simplifies to 1, matching the second equation perfectly.

By verifying both equations, we ensure that the solution is consistent and accurate, confirming the correctness of the process used to solve the system. When errors are found, revisit each step to pinpoint and correct any mistakes.