Problem 47 - Understanding Elementary Algebra with Geometry

Question

In Exercises \(29-62,\) solve the system of equations using any method you choose. $$\left\\{\begin{array}{l} 5 x-3 y=20 \\ 7 x+2 y=28 \end{array}\right.$$

Step-by-Step Solution

Answer

The solution is \(x = 4\) and \(y = 0\).

Step 1 - Choose the method

We'll use the substitution method to solve the system of equations.

Step 2 - Solve one equation for one variable

Solve the first equation for one of the variables. Let's solve for x.The first equation is: \[5x - 3y = 20\]Rearrange to get x:\[5x = 20 + 3y\]\[x = \frac{20 + 3y}{5}\]

Step 3 - Substitute into the second equation

Substitute \(x = \frac{20 + 3y}{5}\) into the second equation.The second equation is:\[7x + 2y = 28\]Substitute x:\[7\left(\frac{20 + 3y}{5}\right) + 2y = 28\]

Step 4 - Simplify the equation

Simplify and solve for y:\[7 \cdot \frac{20 + 3y}{5} + 2y = 28\]\[\frac{140 + 21y}{5} + 2y = 28\]Multiply both sides by 5 to clear the fraction:\[140 + 21y + 10y = 140\]\[31y = 0\]\[y = 0\]

Step 5 - Solve for x

Substitute y back into the expression for x:\[x = \frac{20 + 3(0)}{5}\]\[x = 4\]

Step 6 - Check the solution

Substitute \(x = 4\) and \(y = 0\) back into the original equations to ensure they satisfy both equations:First equation:\[5x - 3y = 20\]\[5(4) - 3(0) = 20\]\[20 = 20\]Second equation:\[7x + 2y = 28\]\[7(4) + 2(0) = 28\]\[28 = 28\]Both equations are satisfied.

Key Concepts

substitution method

The substitution method is a powerful tool for solving systems of equations. It involves solving one of the equations for one variable and then substituting this expression into the other equation. This reduces the system to a single equation with one variable, making it easier to solve.

solving equations

Solving equations is the process of finding the value(s) of the variable(s) that make the equation true. In the context of systems of equations, we are looking for a common solution that satisfies all equations in the system.

algebraic manipulation

Algebraic manipulation involves rearranging equations and expressions using algebraic operations such as addition, subtraction, multiplication, and division. This skill is essential when using methods like substitution to isolate variables and solve systems of equations.

checking solutions

Checking solutions ensures that the values obtained for the variables actually satisfy all original equations. This step is crucial to confirm the accuracy of the solution. By substituting the values back into the original equations, we can verify that each one holds true.