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# 7.4: Solving Linear Systems by Multiplication

Difficulty Level: At Grade Created by: CK-12

This chapter has provided three methods to solve systems: graphing, substitution, and elimination through addition and subtraction. As stated in each lesson, these methods have strengths and weaknesses. Below is a summary.

Graphing

\begin{align*}\checkmark\end{align*} A good technique to visualize the equations and when both equations are in slope-intercept form.

• Solving a system by graphing is often imprecise and will not provide exact solutions.

Substitution

\begin{align*}\checkmark\end{align*} Works particularly well when one equation is in standard form and the second equation is in slope-intercept form.

\begin{align*}\checkmark\end{align*} Gives exact answers.

• Can be difficult to use substitution when both equations are in standard form.

\begin{align*}\checkmark\end{align*} Works well when both equations are in standard form and the coefficients of one variable are additive inverses.

\begin{align*}\checkmark\end{align*} Answers will be exact.

• Can be difficult to use if one equation is in standard form and the other is in slope-intercept form.
• Addition or subtraction does not work if the coefficients of one variable are not additive inverses.

Although elimination by only addition and subtraction does not work without additive inverses, you can use the Multiplication Property of Equality and the Distributive Property to create additive inverses.

Multiplication Property and Distributive Property:

If \begin{align*}ax+by=c\end{align*}, then \begin{align*}m(ax+by)=m(c)\end{align*} and \begin{align*}m(ax+by)=m(c)\rightarrow(am)x+(bm)y=mc\end{align*}

While this definition box may seem complicated, it really states you can multiply the entire equation by a particular value and then use the Distributive Property to simplify. The value you are multiplying is called a scalar.

Example: Solve the system \begin{align*}\begin{cases} 7x+4y=12\\ 5x-2y=11 \end{cases}\end{align*}.

Solution: Neither variable has additive inverse coefficients. Therefore, simply adding or subtracting the two equations will not cancel either variable. However, there is a relationship between the coefficients of the \begin{align*}y-\end{align*}variable.

\begin{align*}4 \ is \ the \ additive \ inverse \ of-2 \times (2)\end{align*}.

By multiplying the second equation by the scalar 2, you will create additive inverses of \begin{align*}y\end{align*}. You can then add the equations.

\begin{align*}\begin{cases} 7x+4y=12\\ 2(5x-2y=11) \end{cases} & \rightarrow \quad \begin{cases} 7x+4y=12\\ 10x-4y=22 \end{cases}\end{align*}

\begin{align*}\text{Add the two equations.} && 17x&=34\\ \text{Divide by} \ 17. && x&=2\end{align*}

To find the \begin{align*}y-\end{align*}value, use the Substitution Property in either equation.

\begin{align*}7(2)+4y&=12\\ 14+4y&=12\\ 4y&=-2\\ y&=-\frac{1}{2}\end{align*}

The solution to this system is \begin{align*}\left ( 2, -\frac{1}{2} \right )\end{align*}.

Example: Andrew and Anne both use the I-Haul truck rental company to move their belongings from home to the dorm rooms on the University of Chicago campus. I-Haul has a charge per day and an additional charge per mile. Andrew travels from San Diego, California, a distance of 2,060 miles in five days. Anne travels 880 miles from Norfolk, Virginia, and it takes her three days. If Anne pays $840 and Andrew pays$1,845.00, what does I-Haul charge:

a) per day?

b) per mile traveled?

Solution: Begin by writing a system of linear equations: one to represent Anne and the second to represent Andrew. Let \begin{align*}x=\end{align*} amount charged per day and \begin{align*}y=\end{align*} amount charged per mile.

\begin{align*}\begin{cases} 3x+880y=840\\ 5x+2060y=1845 \end{cases}\end{align*}

There are no relationships seen between the coefficients of the variables. Instead of multiplying one equation by a scalar, we must multiply both equations by the least common multiple.

The least common multiple is the smallest value that is divisible by two or more quantities without a remainder.

Suppose we wanted to eliminate the variable \begin{align*}x\end{align*} because the numbers are smaller to work with. The coefficients of \begin{align*}x\end{align*} must be additive inverses of the least common multiple.

\begin{align*}LCM \ of \ 3 \ and \ 5=15\end{align*}

\begin{align*}\begin{cases} -5(3x+880y=840)\\ 3(5x+2060y=1845) \end{cases} & \rightarrow \quad \begin{cases} -15x-4400y=-4200\\ 15x+6180y=5535 \end{cases}\end{align*}

\begin{align*}&\text{Adding the two equations yields:} && 1780y =1335\\ &\text{Divide by} 1780 && \qquad \quad \ y=0.75\end{align*}

11. A baker sells plain cakes for $7 or decorated cakes for$11. On a busy Saturday, the baker started with 120 cakes, and sold all but three. His takings for the day were 991. How many plain cakes did he sell that day, and how many were decorated before they were sold? 12. Twice John’s age plus five times Claire’s age is 204. Nine times John’s age minus three times Claire’s age is also 204. How old are John and Claire? Mixed Review 1. Baxter the golden retriever is lying in the sun. He casts a shadow of 3 feet. The doghouse he is next to is 3 feet tall and casts an 8-foot shadow. What is Baxter's height? 2. A botanist watched the growth of a lily. At 3 weeks, the lily was 4 inches tall. Four weeks later, the lily was 21 inches tall. Assuming this relationship is linear: 1. Write an equation to show the grow pattern of this plant. 2. How tall was the lily at the 5.5-week mark? 3. Is there a restriction on how high the plant will grow? Does your equation show this? 3. The “Wave” is an exciting pasttime at football games. To prepare, students in a math class took the data in the table below. 1. Find a linear regression equation for this data. Use this model to estimate the number of seconds it will take for 18 students to complete a round of the wave. 2. Use the method on interpolation to determine the amount of time it would take 18 students to complete the wave. \begin{align*}&s \ (\text{number of students in wave})&& 4 && 8 && 12 && 16 && 20 && 24 && 28 && 30\\ & t \ (\text{time in seconds to complete on full round}) && 2 && 3.2 && 4 && 5.6 && 7 && 7.9 && 8.6 && 9.1\end{align*} ## Quick Quiz 1. Is (–3, –5) a solution to the system \begin{align*}\begin{cases} -3y=3x+6\\ y=-3x+4 \end{cases}\end{align*}? 2. Solve the system: \begin{align*}\begin{cases} y=6x+17\\ y=7x+20 \end{cases}\end{align*}. 3. Joann and Phyllis each improved their flower gardens by planting daisies and carnations. Joann bought 10 daisies and 4 carnations and paid52.66. Phyllis bought 3 daisies and 6 carnations and paid $43.11 How much is each daisy? How much is each carnation? 4. Terry’s Rental charges$49 per day and $0.15 per mile to rent a car. Hurry-It-Up charges a flat fee of$84 per day to rent a car. Write these two companies' charges in equation form and use the system to determine at what mileage the two companies will charge the same for a one-day rental.

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