Question

Solve each system of equations by addition-subtraction, or by substitution. Check some by graphing. $$\begin{aligned} &5.66 p+4.17 q-16.9=0\\\ &13.7 p=32.2+3.61 q \end{aligned}$$

Short answer

The solution contains values for p and q that satisfy both equations when they are each set to zero. Check this solution by graphing the two lines represented by the equations to see if they intersect at the calculated (p, q) coordinates.

Step-by-step solution

Solve the second equation for p

Take the second equation and isolate the variable p by dividing both sides of the equation by the coefficient of p, which is 13.7. This will give you an expression for p in terms of q.

Substitute the expression for p into the first equation

Once you have p expressed in terms of q, substitute this expression into the first equation wherever you see p. This will give you an equation in one variable, q.

Solve for q

With the substituted first equation now only in terms of q, solve for q by combining like terms and isolating q on one side of the equation.

Solve for p

After finding the value of q, substitute it back into the expression obtained for p in terms of q to solve for the value of p.

Check the solution by graphing

Plot the two original equations on a graph to ensure that the point of intersection corresponds to the solution (p, q) calculated algebraically. If the point where the lines intersect matches the solution found, then the solution is verified.

Key concepts

Addition-Subtraction Method

Understanding how to tackle systems of equations begins with grasping the addition-subtraction method, a powerful technique to eliminate one variable by combining equations. To apply this method, one must align the equations so that adding or subtracting them cancels out one of the variables.

For example, if we multiply the first equation by a number that will make the coefficient of one variable equal and opposite to the coefficient of the same variable in the second equation, we can eliminate that variable by adding or subtracting the two equations. This method is particularly useful when the coefficients are already set up for easy elimination; otherwise, it requires some multiplication or division to prepare the equations for the addition or subtraction step.

Substitution Method

Another popular strategy is the substitution method. It comes into play when we can easily express one variable in terms of another, which then allows us to replace ('substitute') this expression into the other equation. This converts the system of equations into a single equation with one variable.

In the given problem, the second equation was manipulated to isolate variable 'p', yielding an expression that we could substitute into the first equation. Substitution is often the method of choice when we can quickly isolate a variable and when dealing with equations where addition-subtraction might be cumbersome due to awkward coefficients.

Isolate Variables

The crucial step in many algebraic methods, including substitution, involves the ability to isolate variables. This process generally involves rearranging the equation to get the targeted variable by itself on one side of the equal sign while keeping the other variables and constants on the other side.

This is achieved by performing operations such as addition, subtraction, multiplication, and division, but doing so equally on both sides of the equation to maintain balance. In the exercise, we isolated 'p' by dividing both sides by its coefficient. Isolating variables not only prepares for substitution but also simplifies the complexity of the problem.

Graphical Solution Verification

A graphical check of our solution is an excellent verification tool. By plotting each equation on the same set of axes, we can determine if our algebraic solution is correct based on where the lines intersect the graph.

The point of intersection should represent the solution to the system of equations. In our exercise, after solving algebraically, we could plot the original two equations. If done correctly, they should intersect at the point equivalent to our values for 'p' and 'q'. This graphical solution verification acts as a tangible proof that we've solved the system correctly. Moreover, it offers an intuitive understanding of the solution space and the behaviour of the system.