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Chapter 4: Problem 71

Explain how to solve a system of equations using the addition method. Use \(3 x+5 y=-2\) and \(2 x+3 y=0\) to illustrate your explanation.

Short Answer

Expert verified
The solution to the system of equations is \(x = 6, y = -4\).

Step by step solution

01

Identify the Variable to Eliminate

The easiest variable to eliminate is y because multiplying the first equation by 3 and the second by -5 gives coefficients for y that are equal and opposite. Thus, multiplying the first equation by 3 gives 9x + 15y = -6, and multiplying the second equation by -5 gives -10x - 15y = 0.
02

Add The Equations Together

The addition of the two equations from step 1 yields an equation with only one variable. This is because the y terms cancel out when summed. Therefore, (9x + 15y) + (-10x - 15y) equals -1x (or -x) and -6 + 0 equals -6. This provides us with the equation, -x = -6.
03

Solve for the Remaining Variable

Once we have the equation -x = -6, we can solve for x by multiplying both sides of the equation by -1. Therefore, x = 6.
04

Substitute the value of x into one of the original equations

We can substitute x = 6 into the original equation 2x + 3y = 0 to find the value of y. This gives: 2(6) + 3y = 0, which simplifies to 12 + 3y = 0, and then, 3y = -12. Therefore, y = -12 / 3, which gives us y=-4.
05

Checking the solution

The pair of value (x , y) = (6 , -4) should satisfy both the equation of the system. Substituting these values into the two original equations confirms that the left-hand side is equal to the right-hand side, thus validating the solution.

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Most popular questions from this chapter

The law of supply and demand states that, in a free market economy, a commodity tends to be sold at its equilibrium price. At this price, the amount that the seller will supply is the same amount that the consumer will buy. Explain how systems of equations can be used to determine the equilibrium price.

Solve each system or state that the system is inconsistent or dependent. $$\left\\{\begin{array}{l}0.4 x+y=2.2 \\ 0.5 x-1.2 y=0.3\end{array}\right.$$

Describe what happens when using algebraic methods to solve an inconsistent system.

Use the four-step strategy to solve each problem. Use \(x, y,\) and \(z\) to represent unknown quantities. Then translate from the verbal conditions of the problem to a system of three equations in three variables. A certain brand of razor blades comes in packages of \(6,12,\) and 24 blades, costing \(\$ 2, \$ 3,\) and \(\$ 4\) per package, respectively. A store sold 12 packages containing a total of 162 razor blades and took in \(\$ 35 .\) How many packages of each type were sold?

Solve each system or state that the system is inconsistent or dependent. $$\left\\{\begin{array}{l}\frac{x}{2}=\frac{y+8}{3} \\\ \frac{x+2}{2}=\frac{y+11}{3}\end{array}\right.$$
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