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Category: Heart of Algebra—systems of linear inequalities

Here’s how to solve it:

In this question, we’re told coordinates (a,b) lies in the solution set of these equations, and we want to know a maximum possible value. With inequalities, the graph will be shaded to include the set of values that satisfy the inequality, so (a,b) lies in this overlapping region. Because it’s a grid-in response (only real numbers can be the answer to the grid-in questions—there is no option to bubble in infinity) we know that the value of b must be limited by the point of intersection. Since the inequalities are both less than or equal to, we know that the lines (and the point of intersection) are included in the solution set. If there were only less/greater than symbols, the lines would not be included in the solution set.

• This means we can find the point of intersection to find a maximum value of b and solve it just like a system of equations using the substitution method. This gives us the inequality 5x≤−15x+3000. In this case, x is equivalent to a, and the y-value is equivalent to b.

• Move the variables to the same side by adding 15x to both sides. 20x≤3000.

• Divide both sides by 20 to find out what x is. x≤150. Many students might stop here, but remember we want coordinate b in (a,b), or the y-value.

• We can plug x into either equation to find the value of b. We picked the second one since it’s simpler: 5⋅150=750. That’s our answer!

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