Problem 21 - A Graphical Approach to College Algebra

Question

Solve each system by substitution. $$\begin{aligned}&4 x+5 y=7\\\&9 y=31+2 x\end{aligned}$$

Step-by-Step Solution

Answer

The solution is \( x = -2 \) and \( y = 3 \).

Step 1: Solve one equation for one variable

Start by solving the second equation for \( y \): \[ 9y = 31 + 2x \]Divide every term by 9 to isolate \( y \): \[ y = \frac{31 + 2x}{9} \]

Step 2: Substitute into the first equation

Substitute the expression for \( y \) from Step 1 into the first equation:\[ 4x + 5\left(\frac{31 + 2x}{9}\right) = 7 \]

Step 3: Clear the fraction

Multiply every term by 9 to eliminate the fraction:\[ 9(4x) + 5(31 + 2x) = 7 \times 9 \]This simplifies to:\[ 36x + 5(31 + 2x) = 63 \]

Step 4: Distribute and combine like terms

Distribute the 5 into the parenthesis:\[ 36x + 155 + 10x = 63 \]Combine like terms:\[ 46x + 155 = 63 \]

Step 5: Solve for \( x \)

Subtract 155 from both sides:\[ 46x = 63 - 155 \]\[ 46x = -92 \]Divide by 46:\[ x = -2 \]

Step 6: Substitute back to find \( y \)

Use the expression for \( y \) from Step 1 and substitute \( x = -2 \):\[ y = \frac{31 + 2(-2)}{9} \]Calculate:\[ y = \frac{31 - 4}{9} \]\[ y = \frac{27}{9} \]\[ y = 3 \]

Step 7: Verify the solution

Substitute \( x = -2 \) and \( y = 3 \) back into the original equations to verify:For the first equation: \[ 4(-2) + 5(3) = -8 + 15 = 7 \] For the second equation:\[ 9(3) = 31 + 2(-2) = 31 - 4 = 27 \] Both equations are satisfied, so the solution is correct.

Key Concepts

System of Equations

In algebra, a system of equations refers to more than one equation that is solved together. There are often several unknowns or variables involved. When you solve a system of equations, you're looking to find a set of values for the variables, which make all equations true simultaneously. For instance, in our example, the system is:
- \(4x + 5y = 7\)- \(9y = 31 + 2x\)
To solve such systems, methods like substitution, elimination, or graphical methods can be used. Solving systems is akin to finding where lines or curves intersect on a graph, which is why understanding the graphical interpretation can be helpful. But for systematic accuracy, algebraic techniques are preferred.

Solving Equations

Solving equations involves finding the value of the variable that makes the equation true. In the context of systems of equations, it often involves isolating one variable and substituting it into another equation. This process begins by solving one equation for one of the variables, making it easy to substitute.
Continuing with our example, we solved for \(y\) in terms of \(x\) using the second equation:
\[ y = \frac{31 + 2x}{9} \]
This expression for \(y\) is then substituted back into the first equation, allowing us to solve for \(x\) effectively. This substitution method is particularly useful when equations are linear and can simplify other algebraic operations.

Linear Equations

Linear equations are mathematical expressions where each term is either a constant or the product of a constant and a single variable. The highest exponent of variables in linear equations is one, making them straightforward. For example, in our system:
- \(4x + 5y = 7\)- \(9y = 31 + 2x\)
Both equations have linear terms. Linear equations have a graph that is a straight line, and solving them involves finding points of intersection — the solution to the system. Methods like substitution are well suited for solving linear equations because they simplify substitution and elimination of variables.

Verification of Solutions

After obtaining potential solutions for a system of equations, always verify to ensure accuracy. This confirms that the solution satisfies all original equations. Verification involves substituting the solved values back into the original equations and checking for consistency.
For example, with \(x = -2\) and \(y = 3\), we substitute back:
- First equation: \(4(-2) + 5(3) = 7\) - Calculate: \(-8 + 15 = 7\), which is correct.
- Second equation: \(9(3)\) - Check: \(31 + 2(-2) = 27\) - Calculate: \(31 - 4 = 27\), which matches the left side.
These verifications confirm the solution is accurate, ensuring consistency in findings. Always perform this step to catch potential arithmetic errors early.