The local farmer's market sells a 4-pound basket of apples which contains 10

In grade 8, students learned that some systems of linear equations have many solutions. This warm-up reminds students about this fact, while also prompting them to use what they learned in this unit to better understand what it means for a system to have infinitely many solutions.

The first equation shows two variables adding up to 3, so students choose a pair of values whose sum is 3. They notice that all pairs chosen are solutions to the system. Next, they try to find a strategy that can show that there are countless other pairs that also satisfy the constraints in the system. Monitor for these likely strategies:

Solving by graphing: The graphs of the two equations are the same line, so all the points on the line are solutions to the system.

Solving by substitution: The first equation can be rearranged to . Substituting for in the second equation gives or or . This equation is true no matter what is.

Solving by elimination: If we multiply the first equation by 4, rearrange the second equation to , and then subtract the second equation from the first, the result is . Subtracting from each side gives , which is true regardless of what or is.

Reasoning about equivalent equations: If we rearrange the second equation so that the variables are on the same side , we can see that this equation is a multiple of the first and are equivalent. This means they have the exact same solution set, which contains infinite possible pairs of and .

Identify students using different strategies and ask them to share their thinking with the class later.

Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).

Step-by-step explanation:

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7 0

3 years ago

Can someone help me with this? Please this would mean so much.

Paha777 [63]

Answers:

x = angle MLN = 35 degrees

angle FLJ = 55 degrees

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Explanation:

This problem is fairly tricky if you're not sure what to look for. A slight clue is that they've marked a red point that strangely doesn't have a label on it. It's a fairly small point but it's definitely there if you look closely. While this point's location isn't exactly what we want, we're fairly close. Start at point L and draw a ray through the center point P. Ray LP will intersect the circle at point A, as shown in the diagram below.

From here, draw segments FA and MA. Now notice that inscribed angle LMF = 55 and inscribed angle FAL both subtend the same arc. The term "subtend" basically means "cut off". This arc that the inscribed angles subtend is minor arc FL. A minor arc is where you travel the shorter path around the circle, which indicates its measure is less than 180 degrees.

Since inscribed angles LMF and FAL subtend the same minor arc, this makes the inscribed angles to be congruent.

In short: angle FAL is 55 degrees

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Segment LA goes through the center P. Through Thales theorem, we know that inscribed angle LFA is 90 degrees. Consequently, we can determine that inscribed angle FLA is 90-55 = 35 degrees.

Segment JL is tangent to the circle, meaning that angle ALJ is 90 degrees. So angle FLJ is 90-35 = 55 degrees.

It's not a coincidence that angle FLJ, angle FAL, and angle LMF are the same measure.

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We found that angle FAL was 55 degrees. Applying Thales theorem again shows that angle MAF is 90 degrees. Therefore, angle LAM is 90-55 = 35 degrees.

Focus now on triangle LMA. This is also a right triangle (Thales Theorem). The upper acute angle we found was angle LAM = 35, so the lower acute angle is ALM = 55.

Then we can find angle MLN = (angle ALN) - (angle ALM) = 90 - 55 = 35

In short, angle MLN = 35 degrees

Similar to the previous section, it is not a coincidence that angles LFM, LAM and MLN are the same measure.

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As an alternative, since we know angle FLJ = 55, and angle FLM is 90 degrees, this means...

(angle FLJ)+(angle FLM) + (angle MLN) = 180

55 + 90 + angle MLN = 180

145 + angle MLN = 180

angle MLN = 180 - 145

angle MLN = 35 degrees

7 0

2 years ago

Help me wick show you work

expeople1 [14]

3 to 9 clouds are smiley faces Explaintion- hope it’s right

3 0

2 years ago

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