Solving Linear Systems

To begin, we'll use variables to represent the different quantities. Let $w$ be the number of points the Wombats scored and $s$ be the number of points the Seagulls scored. The Wombats scored $13$ more points than the Seagulls. Thus, the difference between $w$ and $s$ can be written as $$w=s+13.$$ The total amount of points was $41,$ so the sum of $w$ and $s$ is $$w+s=41.$$ Both of these equations must be true simultaneously, giving us the following system of equations. $$\{\begin{array}{l}w=s+13\\ w+s=41\end{array}$$ We can solve the system by graphing. First, let's write the second equation in slope-intercept form by subtracting $s$ on both sides. $$\{\begin{array}{l}w=s+13\\ w=\text{-}s+41\end{array}$$ Now, we can graph the lines. Since the scores cannot be negative, we only graph the lines for positive values of $s$ and $w.$

Now, we can identify the point of intersection.

The point of intersection is $(14,27).$ This means, the Wombats scored $27$ points and the Seagulls scored $14.$

We can use the given information to write two equations. First, we must define our variables. Let the first number be $x$ and the other $y.$ We know that the sum of these numbers is $17.$ Thus, $$x+y=17.$$ We also know that one of the numbers, let's say $x,$ is two more than three times the other number, which is then $y.$ This gives us the equation $$x=3y+2.$$ Together these two equations create the system $$\{\begin{array}{l}x+y=17\\ x=3y+2.\end{array}$$ To solve this system using substitution, we must substitute one equation into the other. Let's substitute $x=3y+2$ into $x+y=17.$ This will allow us to then solve for $y.$

We've found that the $y$-coordinate of the solution is $y=3.75.$ We can substitute this value into either equation to solve for the corresponding $x$-value. We'll choose the second equation. Since $x=13.25,$ the solution to the system is $(13.25,3.75).$

There are a number of conditions in this exercise. Let's start with making sense of them. Let's use $b$ to represent the number of burritos that Marco buys, and $t$ to represent the number of tacos. Since the burritos cost $\$5$ each, the total amount spent on just burriots is $5b.$ Similarly, since a taco costs $\$3,$ the total cost of tacos is $3t.$ Therefore, The total cost can be expressed as $$5b+3t.$$ Since Marco has $\$90,$ he cannot spend more than this. The total must be less than or equal to $90.$ That means, $$5b+3t\le 90.$$ We also know that Marco needs to buy enough food to feed $10$ people. Assuming no one wants to share a taco or a burrito, the total amount of dishes must be at least $10.$ This gives $$b+t\ge 10.$$ Since both of these inequalities must hold true, we get the following system of inequalities. $$\{\begin{array}{l}5b+3t\le 90\\ b+t\ge 10\end{array}$$ Solving this system gives all of the possible combinations of burritos and tacos Marco can purchase while buying enough food and staying within his budget. To solve the system, we must graph it. Let's first write the inequalities in slope-intercept form by isolating $b.$ The first inequality can be expressed as $b\le \text{-}0.6t+18.$ Writing the second inequality in slope-intercept form can be done in one step. Specifically, by subtracting $t$ on both sides. $$b+t\ge 10\iff b\ge \text{-}t+10$$ The system can now be expressed as $$\{\begin{array}{l}b\le \text{-}0.6t+18\\ b\ge \text{-}t+10.\end{array}$$ We'll graph each inequality by showing its boundary line and shading the appropriate region.

The purple region represents the solution set that satisfies both inequalities. Since Marco can't buy a negative number of burritos or tacos we're only interested in the positive values of $b$ and $t.$

Any point in this region corresponds to a combination on burritos and tacos that costs less than $\$90$ and feeds at least $10$ people. Let's look at the corners of this region.

The marked points represent minimum and maximum possibilities.

• $(10,0)$ and $(0,10)$ represent Marco buying either $10$ tacos or $10$ burritos. Then he'd feed exactly $10$ people, and have money remaining.

• $(0,18)$ and $(30,0)$ tell how many of each dish Marco can buy if he used all money on and only bought tacos or burritos. Thus, he can buy $18$ burritos or $30$ tacos.

Since we don't know how hungry the guests are, or what their preferences are, Marco should buy both burritos and tacos. Let's choose a point in the middle of the region.

One possibility is that Marco can purchase $12$ tacos and $8$ burritos. That way, there will probably be enough food, and he'll have money left. Note that even though decimal numbers are a part of the solution set, the answer should be given in whole numbers, assuming you can't buy a part of a taco or burrito.