Chapter 9: Problem 10
Solve each system of equations by addition-subtraction, or by substitution. Check some by graphing. $$\begin{aligned} &2 x-3 y=5\\\ &3 x+3 y=10 \end{aligned}$$
Short Answer
Expert verified
The solution to the system of equations is (x, y) = (4, 2).
Step by step solution
01
Simplify the second equation
The second equation can be simplified by dividing both sides by 3 to get rid of the common factor, making the system easier to solve. Divide each term of the second equation by 3: \(3x + 3y = 10\) becomes \(x + y = \frac{10}{3}\).
02
Isolate a variable from the simplified equation
Isolate one variable in the simplified equation from Step 1. We will isolate x: \(x + y = \frac{10}{3}\) becomes \(x = \frac{10}{3} - y\).
03
Substitute the isolated variable in the first equation
Substitute the expression of x from Step 2 into the first equation to find the value of y: Replace x in \(2x - 3y = 5\) with \(\frac{10}{3} - y\) to get \(2\left(\frac{10}{3}-y\right) - 3y = 5\).
04
Solve for y
Solve the resulting equation from Step 3 to find the value of y: Distribute 2 in the left side of the equation and combine like terms: \(\frac{20}{3} - 2y - 3y = 5\) simplifies to \(\frac{20}{3} - 5y = 5\). Finally, solve for y.
05
Find the value of x
Once y is found, substitute it back into the equation \(x = \frac{10}{3} - y\) to find the value of x.
06
Verify the solution by substitution into both original equations
Substitute the found x and y values back into both original equations to verify if they satisfy both equations.
07
Check the solution by graphing (optional)
Graph both equations on the coordinate plane and ensure that their intersection point is the solution point (x, y) found in the previous steps.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Addition-Subtraction Method
The addition-subtraction method is a practical technique for solving systems of linear equations. It involves manipulating the given equations so that when they are added or subtracted, one variable is eliminated, making it possible to solve for the remaining variable.
In our example, the addition-subtraction method can be used by multiplying one or both of the equations by suitable numbers to get coefficients of y that are opposites. For instance, if you multiply the second equation \(3x + 3y = 10\) by -1 after simplifying, it would yield \(x + y = \frac{10}{3}\) becoming \(x + y = \frac{10}{3}\) and \( - x - y = -\frac{10}{3}\). When these two new equations are added together, the y terms cancel out:
In our example, the addition-subtraction method can be used by multiplying one or both of the equations by suitable numbers to get coefficients of y that are opposites. For instance, if you multiply the second equation \(3x + 3y = 10\) by -1 after simplifying, it would yield \(x + y = \frac{10}{3}\) becoming \(x + y = \frac{10}{3}\) and \( - x - y = -\frac{10}{3}\). When these two new equations are added together, the y terms cancel out:
- \((2x - 3y) + (-x - y) = 5 - \frac{10}{3}\)
- This simplification leads to \(x - 4y = \frac{5}{3}\).
Substitution Method
The substitution method is like a game of dominoes, where knocking down one piece leads to the next falling into place. This method involves expressing one variable in terms of the other, and then substituting this expression back into the other equation.
In our simplified system, after dividing the second equation by 3, we isolated x in terms of y from the equation \(x + y = \frac{10}{3}\), which gave us \(x = \frac{10}{3} - y\). This expression of x is then substituted into the first equation, as shown in steps 2 and 3 of the provided solution. By substituting, we convert the two-variable system into a single-variable equation, making it simpler to solve.
In our simplified system, after dividing the second equation by 3, we isolated x in terms of y from the equation \(x + y = \frac{10}{3}\), which gave us \(x = \frac{10}{3} - y\). This expression of x is then substituted into the first equation, as shown in steps 2 and 3 of the provided solution. By substituting, we convert the two-variable system into a single-variable equation, making it simpler to solve.
- The substitution reduces the system to \(2(\frac{10}{3} - y) - 3y = 5\), which then needs to be simplified to solve for y.
- After finding the value of y, it is substituted back into the isolated x equation to find the corresponding x value.
Graphical Method
The graphical method turns an algebraic problem into a visual one. Here, both equations in the system are graphed on the same set of axes, and the point where they intersect represents the solution to the system—the values of x and y that satisfy both equations simultaneously.
To graph the equations from our example, we first convert each to slope-intercept form (y = mx + b). The first equation \(2x - 3y = 5\) can be rearranged to \(y = \frac{2}{3}x - \frac{5}{3}\), and the second one \(x + y = \frac{10}{3}\) becomes \(y = -x + \frac{10}{3}\).
To graph the equations from our example, we first convert each to slope-intercept form (y = mx + b). The first equation \(2x - 3y = 5\) can be rearranged to \(y = \frac{2}{3}x - \frac{5}{3}\), and the second one \(x + y = \frac{10}{3}\) becomes \(y = -x + \frac{10}{3}\).
- Plot both lines on a graph; the first line rises with a positive slope, while the second line falls with a negative slope.
- The intersection point of these lines is the solution to the system.