Problem 152 - Elementary Algebra

Question

In the following exercises, solve the systems of equations by substitution. $$ \left\\{\begin{array}{l} 2 x+9 y=-4 \\ 3 x+13 y=-7 \end{array}\right. $$

Step-by-Step Solution

Answer

x = -11, y = 2

Step 1 - Solve one equation for one variable

Start by solving the first equation for one of the variables. Let's solve for x: divide both sides by 2:\[ x = -\frac{4 + 9y}{2} \]

Step 2 - Substitute into the second equation

Next, substitute the expression for x from Step 1 into the second equation:\[ 3 \left( -\frac{4 + 9y}{2} \right) + 13y = -7 \]

Step 3 - Simplify the equation

Simplify the equation from Step 2 to solve for y:\[ \frac{-12 - 27y}{2} + 13y = -7 \]Multiply through by 2 to eliminate the fraction:\[ - 12 - 27y + 26y = -14 \]Combine like terms:\[ - 12 - y = - 14 \]Add 12 to both sides:\[ - y = - 2 \]Therefore, \[ y = 2 \]

Step 4 - Substitute value of y back into one of the original equations

Substitute y = 2 back into the first equation to find x:\[ 2x + 9(2) = -4 \]Simplify and solve for x:\[ 2x + 18 = - 4 \] Subtract 18 from both sides:\[ 2x = -22 \]Divide by 2:\[ x = -11 \]

Step 5 - Write the solution

The solutions for the system of equations are x = -11 and y = 2.

Key Concepts

Solving Systems of Equations

When tackling systems of equations, we are essentially dealing with two or more equations that share common variables. Our goal is to find values for these variables that satisfy all given equations simultaneously. A common method for solving these systems is the substitution method, which is both a systematic and logical approach.
Here’s a brief overview of the substitution method:

Linear Equations

Linear equations are algebraic expressions that connect variables using a straight line when plotted on a graph. These equations typically appear in the first degree, meaning the variables are not squared, cubed, or altered in any other nonlinear way. For example, the equation: 2x + 9y = -4 is a linear equation because the highest degree of x and y is 1.
By examining the structure of linear equations, we can determine the relationships between variables, helping us solve various mathematical problems.

Algebraic Substitution

Algebraic substitution is a technique used to solve equations by substituting one variable with an equivalent expression derived from another equation. This method simplifies complex systems into single-variable equations, making them easier to solve.
Here’s how you can use substitution to solve a system of equations:

• Solve one equation for a particular variable.
• Substitute this expression into the other equation(s).
• Simplify and solve the resulting equation.
• Use the solution from the simplified equation to find the other variable(s).

Let's break it down further:
Step 1: Choose one equation and solve for one variable. For instance, in the equation 2x + 9y = -4, we solve for x: \[ x = -\frac{4 + 9y}{2} \]
Step 2: Substitute this expression into the other equation. This means inserting \[ x = -\frac{4 + 9y}{2} \] into the second equation: 3x + 13y = -7, resulting in: \[ 3 \left( -\frac{4 + 9y}{2} \right) + 13y = -7 \]
Step 3: Simplify and solve for the remaining variable. Often, this step involves combining like terms and potentially eliminating fractions: \[ \frac{-12 - 27y}{2} + 13y = -7 \]
Step 4: Use the newly found variable’s value to find the original variable by substituting it back. In our example, we find y = 2 and then substitute back to find x. Finally, conclude with the solutions for all variables involved. This method ensures that every step is clear and verified, ultimately solving the system efficiently. (Copy and paste relevant part of initial solution to show how it's done in practice)