Problem 15 - A Graphical Approach to College Algebra

Question

Solve each system by substitution. $$\begin{aligned}&y=2 x+3\\\&3 x+4 y=78\end{aligned}$$

Step-by-Step Solution

Answer

The solution is \( x = 6 \), \( y = 15 \).

Step 1: Identify Substitution Equation

Look at the given system of equations. The first equation is already solved for one variable: \( y = 2x + 3 \). This equation will be used for substitution into the second equation.

Step 2: Substitute y in the Second Equation

Substitute \( y = 2x + 3 \) from the first equation into the second equation \( 3x + 4y = 78 \). This gives us: \( 3x + 4(2x + 3) = 78 \).

Step 3: Simplify the Equation

Expand and simplify the equation by distributing \(4\) through \( (2x + 3) \): \( 3x + 8x + 12 = 78 \). Then combine like terms: \( 11x + 12 = 78 \).

Step 4: Solve for x

Isolate \( x \) in the equation \( 11x + 12 = 78 \). Subtract \( 12 \) from both sides: \( 11x = 66 \). Then, divide both sides by \( 11 \): \( x = 6 \).

Step 5: Substitute x Back to Find y

Use the value of \( x \) to find \( y \) using the first equation \( y = 2x + 3 \). Substitute \( x = 6 \) to get: \( y = 2(6) + 3 \). Simplify to find \( y = 12 + 3 = 15 \).

Step 6: Verify the Solution

Substitute \( x = 6 \) and \( y = 15 \) back into the original second equation \( 3x + 4y = 78 \) to verify: \( 3(6) + 4(15) = 18 + 60 = 78 \). Both sides are equal, confirming the solution is correct.

Key Concepts

Substitution Method

The substitution method is a technique for solving a system of equations by replacing one variable in one of the equations with an expression obtained from the other equation. This is particularly useful when one of the equations is already solved for a variable, such as when you have something like \( y = 2x + 3 \). This approach can simplify the problem, making it easier to solve.

• Begin by selecting one of the equations which is solved for a variable. In the given example, the equation \( y = 2x + 3 \) is perfect for substitution.
• This expression for \( y \) will be inserted into the other equation, replacing the \( y \) variable. This reduces the entire system to a single equation with one variable.

Once you perform the substitution and simplify, you solve for the remaining variable. This method ensures that both original equations are taken into account while solving for the unknowns. The substitution method is particularly effective for systems where one equation is easily solvable in terms of one variable.

Linear Equations

Linear equations are fundamental in algebra and represent straight lines when graphed on the coordinate plane. The general form of a linear equation in two variables is \( ax + by = c \).

• Each term in the equation is either a constant or the product of a constant and a single variable.
• The equations in a system, like \( y = 2x + 3 \) and \( 3x + 4y = 78 \), result in straight lines when plotted.

Linear equations can be tackled using various methods, including the substitution method illustrated here. The goal is to find a common solution — a point where the lines intersect, which represents simultaneous solutions to both equations. Because linear equations form straight lines, their solutions can often be directly visualized geometrically, making it easier to understand if you imagine these lines on a graph.

Algebraic Solution

An algebraic solution refers to solving equations and systems of equations using algebraic manipulations and techniques rather than relying on graphical methods. This includes operations like substitution, elimination, and the simplification of expressions.

• In our exercise, algebraic steps include substituting an expression for one variable and simplifying the result to isolate the remaining variable.
• These steps also involve combining like terms and isolating variables by using basic arithmetic operations such as addition, subtraction, multiplication, and division.

Once you obtain a value for one variable, substitute it back into one of the original equations to solve for the other variable. Finally, verify the solution by substituting both values into the original equations, ensuring they satisfy both equations. This careful verification helps confirm that the algebraic solution is correct and accounts for any arithmetic errors that might have occurred during calculations.