Problem 45 - Algebra 2

Question

Solve each system of equations by using either substitution or elimination. $$ \begin{array}{l}{4 x+5 y=7} \\ {3 x-2 y=34}\end{array} $$

Step-by-Step Solution

Answer

The solution is \(x = 8\) and \(y = -5\).

Step 1: Choose a Method

We will use the elimination method to solve this system of equations. This method will allow us to eliminate one of the variables by adding or subtracting the equations.

Step 2: Prepare the Equations for Elimination

The given system is: \(4x + 5y = 7\) and \(3x - 2y = 34\). We need to manipulate both equations so that either \(x\) or \(y\) has the same coefficient in both equations.

Step 3: Eliminate One Variable

Multiply the first equation by 2 and the second equation by 5 to create matching coefficients for \(y\): \(8x + 10y = 14\) and \(15x - 10y = 170\). Adding these two equations eliminates \(y\): \(23x = 184\).

Step 4: Solve for x

Divide both sides of the equation \(23x = 184\) by 23: \(x = 8\).

Step 5: Substitute x into One Equation

Substitute \(x = 8\) back into the first equation \(4x + 5y = 7\): \(4(8) + 5y = 7\), which simplifies to \(32 + 5y = 7\).

Step 6: Solve for y

Move 32 to the other side to get \(5y = 7 - 32\), which simplifies to \(5y = -25\). Divide both sides by 5 to solve for \(y\): \(y = -5\).

Step 7: Check the Solution

Substitute \(x = 8\) and \(y = -5\) into the second original equation \(3x - 2y = 34\): \(3(8) - 2(-5) = 34\), which simplifies to \(24 + 10 = 34\). The solution is correct.

Key Concepts

elimination method

The elimination method is a strategic way to solve systems of linear equations. This technique focuses on removing one variable, making it easier to solve for the remaining one. You first aim to manipulate the equations so that one variable can be canceled out.
To utilize this method, follow these steps:

• Arrange both equations in standard form (i.e., all variable terms and constants on one side).
• Identify a target variable to eliminate. In most cases, it helps to choose the one with easily adjustable coefficients.
• Multiply one or both equations by suitable numbers so that one variable's coefficients are opposites.
• Add or subtract the equations to eliminate the target variable. This results in an equation with a single variable, simplifying the solving process.

Once one variable is eliminated, the system becomes a straightforward algebraic equation with only one unknown, paving the way to a quick solution.

substitution method

The substitution method is another popular approach to solve linear systems. It's particularly useful when one equation is easily solvable for one of the variables.
To use substitution effectively:

• Select one of the equations and solve it for one variable in terms of the other.
• Substitute this expression into the other equation. This substitution creates an equation with only one variable.
• Solve this single-variable equation to find the value of the first variable.
• Substitute the found value back into the modified equation from earlier to solve for the second variable.

The substitution method is very handy in systems where one of the equations is already isolated, or can be easily isolated, making it less intensive than elimination.

linear equations

Linear equations form the backbone of algebra and represent relationships with constant rates of change. They graph as straight lines, characterized by the formula:

\[ ax + by = c \]Here, \(a\) and \(b\) are coefficients, while \(c\) is a constant term. Linear equations can have one variable, like \(ax + b = 0\), or two variables, as seen in systems of equations.
In a system of linear equations, you're looking for the set of values that simultaneously satisfy all equations involved. Graphically, this is the point where the lines intersect.
Linear systems can be consistent (having a single solution), inconsistent (no solution), or dependent (infinitely many solutions, where both equations describe the same line). Understanding these outcomes aids in choosing the right method to solve the system.

solution verification

After finding potential solutions, it's crucial to verify them. Verification ensures accuracy in your solutions and reinforces your understanding of the methods used.
To verify a solution:

• Substitute the obtained values of variables back into the original equations.
• Check if each equation holds true with these values. For example, plugging in \(x = 8\) and \(y = -5\) should satisfy both original equations in our given problem.
• Ensure that the substituted equations simplify correctly. If so, the solution is verified as correct.

Verification is a simple yet powerful step that can catch potential errors during manual calculations and is an essential component in mathematics for ensuring solution reliability.