What we learned in math 10 is how to solve systems of equations. It was pretty interesting to me that we can just get the value of the x and y using two equations. There are two ways to get the solution of a pair of equations, Substitution and Elimination.  Substitution is useful when there is a coefficient which value is 1. You can use substitution if you remember how to isolate something on an equation. We should choose the simpler equation (e.g x – 2y = 3) firstly and express one variable in terms of the other. After this step, we need to substitute the expression from the last step into the equation and solve the single variable equation. Lastly, substitute the solution into the equation in the first step to find the value of the other variable. We can solve every single of systems of equations with substitution, but it might be really complicated if there is not a simple equation, so we learned another way, Elimination. I prefer elimination to substitution because I think it is easy for all students in the class unless they do not know how to multiply. Using the method of elimination is very simple. If necessary, multiply each equation by a constant to obtain coefficients for x (or y) that are identical (except perhaps for the sign) and add or subtract the two equations to eliminate one of the variables. After this, solve the resulting equation to determine the value of one of the values. Lastly, substitute the solution the solution into either of the original equations to determine the value of the other variable. We have learned about those ways, However, sometimes, we cannot get a solution of the systems of equations. We must make sure it has zero solution if the slope is same on the equations, and it has infinite solutions if the equations are same.

For examples,

1. A rectangle is to be drawn with perimeter 64 cm. If the length is to be 14 cm more than the width, determine the area of the rectangle.

• If the width is y and the length is x, the equations are 2x + 2y = 64 and x – y = 14, and the first equation is divided by 2, so the equations are x – y = 14 and x + y = 32 {= (2x + 2y = 64)/2} (it does not really matter to skip dividing steps)

• Substitution 
• I am going to isolate y on x + y = 32. x + y = 32 => y = – x + 32. We can just substitute the value of y on one of those equations. x – (– x + 32) = 14 => 2x – 32 = 14 => 2x = 46 => x = 23

• Elimination 
• I do not need to multiply any numbers on the equation and just add these two equations. (x + y = 32) + (x – y = 14) => 2x = 46 => x = 23.
• We can just substitute the value of x on one of those equations. (23) + y = 32 => y = 32 – 23 => y = 9

• So, the length is 23 cm, and the width is 9 cm.
• The formula of the area of a rectangle is (length) × (width). (23) × (9) = 207

• So, the area of the rectangle is 207 cm²

2.

• The system of equations is 2x – 3y = – 5 and x + y = 5. I got the solutions in two different ways.
• Substitution
• I isolated x on the second equation, so the equation is x = – y + 5, and substitute the equation on x of the first equation. 2(– y + 5) – 3y = – 5. – 2y + 10 – 3y = – 5 => – 2y – 3y = – 5 – 10 => – 5y = – 15 => y = 3

• Elimination 
• The second equation is multiplied by – 2, – 2(x + y = 5) => – 2x – 2y = – 10, and substract by the first equation. (2x – 3y = – 5) + (– 2x – 2y = – 10) => – 5y = – 15 => y = 3

• The last step is to substitute 3 on y. x + y = 5 => x + 3 = 5 => x = 5 – 3 => x = 2.
• So, the solution is (2,3)

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What we learned in week 15 in Math 10 is about a formula to get slopes and kinds of forms. The formula to get the value of a slope is pretty interesting for me. If the coordinates are (x₁, y₁) and (x₂, y₂), the formula is y₁ – y₂ over x₁ – x_2, so it means, we can the value of slope easily if we know ONLY the two coordinates on the line. A slope is the most important part of these units, and the forms are no exception. We learned two new forms in this week excepting Slope-Intercept Form which we already know. One of them is Pretty General Form, easily, the numbers must exist on only one side. It must consist of integers, it means there is NO fractions and decimals. Also, it must contain more positive numbers. For example, there is – x + y – 1 = 0, x – y + 1 is better. Another form is Point-Slope Form. The form is used when we know a coordinate and the slope. If we know the coordinate (x₁, y₂) and the slope is m, the formula of this form is y – y₂ = m(x – x₁).

For example,

1. When the two coordinates of the slope are (3, 7) and (5, – 13), the value of the slope.

• The formula of a slope is y₁ – y₂ over x₁ – x_2.  We can substitute the coordinates on the formula, then, it is – 13 – 7 over 5 – 3. – 20 over 2 can be divided by 2, so it is – 10.
• So, the slope is – 10.

2.

• The slope is 4
• The y-intercept is 2
• The formula of the line is y = 4x + 2, so the formula of Pretty General Form is 4x – y + 2 =0.
• We know a coordinate (1, 6) and the slope 4. The formula is y – y₂ = m(x – x₁). We should substitute the coordinate and slope. So, Point-Slope From is 4(x – 1) = y -6.

What I learned week 14 in math 10 is about a slope.

The slope of a line segment defines and describes a measure of the steepness of the line segment. The most important part to get a value of the slope is a rise and a run. A rise defines the change in vertical height between the endpoint, and a run is a change in horizontal length between the endpoints. It is difficult to memorize with those meaning because my English skills are weaker, so I decide understanding meanings instead of memorizing. Easily, a rise is a connection with y value and a run is a connection with x value. The rise and run are used to make a ratio to get a value of the slope. The ratio is the rise over the run (rise/run). The rise is POSITIVE if we count UP, NEGATIVE if we count DOWN and the run is POSITIVE if we count Right, and NEGATIVE if we count LEFT.

For example,

• a = it counts UP and RIGHT, so its slope is POSITIVE. The rise is 6 and the run is 2, so the fraction is six over two and it can be divided by 2.
• So, the slope is 3.
• b = it counts UP and LEFT, so its slope is NEGATIVE. The rise is – 9 and the run is even 9, so the fraction is nine over nine and it can be divided by 9.
• So, the slope is – 1.
• c = it counts DOWN and RIGHT, so its slope is NEGATIVE. The rise is – 2 and the run is 6, so the fraction is negative two over six and it can be divided 2.
• So, the slope is – 1/3.
• d = it counts DOWN and LEFT, so its slope is POSITIVE. The rise is – 3 and the run is – 5, so the fraction is negative three over negative five it can be divided by – 1.
• So, the slope is 3/5
• e = it counts DOWN and RIGHT, so its slope is NEGATIVE. The rise is – 1 and the run is 8, so the fraction is one over eight and there is no GCF (Greatest Common Factor) which can divide them.
• So, the slope is – 1/8

We started to learn Relations and Functions Unit this week. It might be the most difficult unit for me in math 10.

I learned about an independent variable and a dependent variable firstly. An independent variable is usually called x, and a dependent variable is usually called y or f(x). A dependent variable is changed by an independent variable in a functional formula. We can get x and y-incepts, a domain, and a range using those variables. When we should get a value of the x-intercept, we must substitute 0 for the y (x-intercept, 0), in contrast, how to get a value of the y-intercept is similarly substitute 0 for the x (0, y-intercept). A domain is the set of all of numbers for the independent variable (input x) in the relation, and a range is almost same as a domain excepting to change the independent variable to the dependent variable (output y). A relation is a comparison between two sets of elements and has connections more than one. A function is same as f(x) and has only one connection.

For example,

1. The cost of jellies is related to the weight

• The cost is a dependent variable, to output, a range, and y.
• The weight is an independent variable, to input, a domain, and x.
• It is a function.

2.

• It is a function. f(x) is same a function, and 2/3 x² – 6 is a notation.
• x-intercept is to substitute 0 for y, 0 = 2/3 x² – 6. So, the x-intercept is equal to 3 and -3.
• y-intercept is to substitute 0 for x, y = 2/3 * 0² -6. So, the y-intercept is equal to -6.
• A domain is the set of all of numbers for the independent variable in the relation. So, the domain is {-3 ≤ x ≤ 3, x ∈ R}
• A range is the set of all of numbers for the dependent variable in the relation. So, the range is {-6 ≤ y ≤ 3, y ∈ R}