Problem 84 - Algebra 2

Question

Solve each system of equations by using either substitution or elimination. $$ \begin{array}{l}{10 x-y=13} \\ {3 x-4 y=15}\end{array} $$

Step-by-Step Solution

Answer

The solution to the system is \( x = 1 \) and \( y = -3 \).

Step 1: Choose Method for Solving

In this exercise, we'll use the elimination method to solve the system of equations, as the coefficients of the equations are suitable for this approach.

Step 2: Align and Prepare Equations

The given system of equations is:\[ \begin{array}{r} 10x - y = 13 \ 3x - 4y = 15 \end{array} \]Our goal is to eliminate one variable by making coefficients of a variable equal in magnitude but opposite in sign.

Step 3: Multiply Equations

We'll eliminate \( y \) by making its coefficients opposites. Multiply the first equation by 4 to align with \(-4y\) from the second equation. This gives us:\[ \begin{array}{r} 40x - 4y = 52 \ 3x - 4y = 15 \end{array} \]

Step 4: Subtract Equations

Subtract the second equation from the first to eliminate \( y \):\[(40x - 4y) - (3x - 4y) = 52 - 15\]which simplifies to:\[37x = 37\]

Step 5: Solve for x

Divide both sides by 37 to solve for \( x \):\[x = 1\]

Step 6: Substitute to Find y

Use the value of \( x \) in the first original equation:\[10(1) - y = 13\]Simplify:\[10 - y = 13\]Subtract 10 from both sides:\[-y = 3\]Multiply by -1 to solve for \( y \):\[y = -3\]

Step 7: Check Solution

Substitute \( x = 1 \) and \( y = -3 \) back into the second equation to verify the solution:\[3(1) - 4(-3) = 15\]Simplify:\[3 + 12 = 15\]The solution satisfies both equations.

Key Concepts

Systems of Equations

A system of equations involves two or more equations that share the same set of variables. In our case, we are working with a system consisting of two linear equations. Each equation represents a line, and the solution to the system is the point where these two lines intersect.

When solving systems of equations, we aim to find values for the variables that make all the equations true simultaneously. There are two common methods to solve these systems: substitution and elimination. It's important to understand both as each has situations where it might be more efficient or intuitive to use.

In our problem, the system of equations looks like this:

• 10x - y = 13
• 3x - 4y = 15

Solving Linear Equations

Linear equations are equations where the variables are raised to the first power. Solving a linear equation typically involves isolating the variable on one side of the equation. We often start by simplifying the equation through methods such as combining like terms or using operations like addition, subtraction, multiplication, or division.

For example, once we use the elimination method to find that 37x = 37, we simply divide both sides by 37 to find x = 1. This step of isolating the variable is central to solving linear equations. It's crucial to perform the same operation on both sides to maintain equality.

Each step must logically follow from the last, ensuring that the solution is correctly derived.

Substitution Method

The substitution method is another powerful way to solve systems of equations. Although we used elimination in the original exercise, it's useful to know how substitution works, as it might be more straightforward in some cases.

Here's a quick overview of substitution:

• Solve one of the equations for one variable in terms of the others.
• Substitute this expression into the other equation. This removes one variable and allows you to solve for the remaining variable.
• Once you have a value for one variable, substitute it back into one of the original equations to find the other variable.

Substitution is especially handy when one equation is already solved for a single variable or can be rearranged easily.

Checking Solutions in Algebra

Verifying your solution is an essential step in solving systems of equations. Even if the calculations seem correct, it's always good practice to double-check. This involves substituting the found values back into the original equations to ensure they satisfy all given equations.

For instance, after finding x = 1 and y = -3, we substitute these values back into the second equation 3x - 4y = 15 to check our work:

• Substitute: 3(1) - 4(-3) = 15
• Simplify: 3 + 12 = 15
• Check: 15 = 15, which confirms the solution is correct.

This step ensures that no arithmetic errors were made and that the solution is indeed valid.