Unit 4 Linear Equations and Linear Systems (Family Materials)
Section A Puzzle Problems
This week your student will work on solving linear equations. We can think of a balanced hanger as a metaphor for an equation. An equation says that the expressions on either side have equal value, just like a balanced hanger has equal weights on either side.
![A balanced hanger diagram with 1 square and 2 triangles on the left side and 5 triangles on the right side illustrating the equation a + 2b = 5b](../../../../embeds/42c8104c--8.4.B1.Image.05.png)
![A balanced hanger diagram with a square on the left and 3 triangles on the right illustrating a = 3b.](../../../../embeds/d1d66941--8.4.B1.Image.06.png)
If we have a balanced hanger and add or remove the same amount of weight from each side, the result will still be in balance.
We can do this with equations as well: adding or subtracting the same amount from both sides of an equation keeps the sides equal to each other. For example, if
Here is a task to try with your student:
Elena and Noah work on the equation
Elena:
Noah:
Do you agree with their solutions? Explain or show your reasoning.
Solution:
No, they both have errors in their solutions.
Elena multiplied both sides of the equation by 2 in her first step, but forgot to multiply the
Noah divided both sides by -3 in his last step, but wrote -8 instead of
Section C Systems of Linear Equations
This week your student will work with systems of equations. A system of equations is a set of 2 (or more) equations where the letters represent the same values. For example, say Car A is traveling 75 miles per hour and passes a rest area. The distance in miles it has traveled from the rest area after
We could also answer the question without using a graph. Since we are asking when the
Here is a task to try with your student:
Lin and Diego are biking the same direction on the same path, but start at different times. Diego is riding at a constant speed of 18 miles per hour, so his distance traveled in miles can be represented by
Solution:
To find when Lin and Diego meet, that is, when they have traveled the same total distance, we can set the two equations equal to one another: