Category: Grade 10

Solving systems of linear equations involves finding the values of variables that coordinate with all the given equations on where they will meet. The process typically follows several steps, and there are different methods to achieve a solution. One common method is the substitution method, where one equation is solved for one variable, and then that expression is substituted into the other equations. Another method is the elimination method, which involves manipulating the equations to eliminate one variable by adding or subtracting equations. The goal is to reduce the system to a single equation with one variable, making it easier to find the solution. The purpose of solving systems of linear equations is to determine the points of intersection, if any, between the corresponding lines or planes represented by the equations. These points of intersection, known as the solution set, represent the values of the variables that work with all the equations. The system can have a unique solution, no solution (inconsistent system), or infinitely many solutions (dependent system). The process of solving these systems provides a very good tool for analyzing and understanding complex systems.

Example of Subsitution

4x+2y=17 and x-3y=6 is taken and turned into x=3y+6

You plug that into the other equation.

4(3y+6)+2y=17 -> 12y+24+2y=17 -> subtract 24 on each side and add the y’s -> 14y=-17=-0.5

Y=-0.5

X= 4x+2(-0.5)=17 -> 4x+1=17 subtract one on both sides 4x=16 -> 4x/4=16/4=4 X=4

(4,-0.5)

Example of Elimination

4x+2y=8 and 2x+y=2 -> 2(2x+2=2) now the new system is

4x+2y=8 and 4x+2y=4

The 4xs cancel out when you add it the 2y and 2y add up and 4 and 8. Making it 4y=8 -> 4y/4=8/4 Y=4

Then plug it in

4x+2(4)=8

4x+8=8 subtract 8 from both sides

4x=0 -> 4x/4=0/4 X=0

 

(0,4)

 

In Week 16 of Math 10 we learned an abundant of things.

Solving Systems.

In math, solving systems using substitution and elimination methods helps find where two or more equations intersect. Substitution swaps variables between equations, letting you solve for one variable and substitute its value into the other equation. Elimination combines equations by adding or subtracting them to eliminate a variable, simplifying the system to solve for the remaining variable(s). Both methods offer ways to crack the code of intersecting lines or points where equations meet.

The elimination method in solving a system with two equations involves changing the equations to eliminate one of the variables. Making it easier to solve for the remaining variable. You do this by adding or subtracting the equations in a way that cancels out one of the variables when combined.

For example, if you have two equations with the same variable but opposite coefficients (like 2x and -2x), adding them will eliminate that variable, leaving you with an equation involving only the other variable. By repeating this process with different combinations, you gradually narrow down and solve for both variables, uncovering the precise values where the equations intersect on the coordinate plane.

The substitution method in solving a system with two equations involves isolating one variable in terms of the other from one equation and substituting that expression into the other equation. By doing so, you replace one variable in the second equation with the expression found in the first equation, effectively creating an equation with just one variable. This simplifies the system, allowing you to solve for that variable. Once you find the value of one variable, you can substitute it back into either of the original equations to solve for the other variable, pinpointing the exact intersection point or solution where the equations meet on the coordinate plane.

Example

We have a system of equations:

Equation 1: 4x+3y=15

Equation 2: $2x-y=7$

We’re going to use the substitution method to solve this system. Looking at Equation 2, we can isolate $y$:

$y=2x-7$

Now that we have $y$ in terms of $x$, we substitute this expression into Equation 1:

$4x+3(2x-7)=15$

Now, solve for $x$:

$4x+6x-21=15$

Combine like terms:

$10x-21=15$

Add 21 to both sides:

$10x=36$

Divide both sides by 10:

$x=3.6$

Now that we’ve found $x$, let’s substitute it back into $y=2x-7$ to solve for $y$:

$y=2(3.6)-7$

$y=7.2-7$

$y=0.2$

Therefore, the solution to the system of equations is $x=3.6$ and $y=0.2$.

Purpose is to calculate at points/coordinates where two lines will cross. Zero times, One time, Infinite times.

Graphing

You can have two equations to solve a graph.

2x-10=y Change it to y-intercept form

y=2x-10

Starts at 0,-10 and rises 2 right by 1.

Second equation

-1/3(x+5)=y-1 these coordinates are (-5,1).

Then this line crosses the second at (4,-2)

-1/3(4+5)=-2-1

-1/3(9)=-3

-3=-3.

Rise Over Run

“Rise over run” is a term used in math to describe the slope of a line. It refers to the vertical change (rise) divided by the horizontal change (run) between two points on a line. Using an coordinate system, for example, if you have two points (x1, y1) and (x2, y2), the rise over run formula to find the slope (m) of the line passing through these points. You can tell if a slope is positive by just looking at a graph.

Variables (X,Y)

X and Y are variables in math that are used to represent unknown or changing quantities. They are frequently employed in algebraic expressions, where the independent variable in a function or equation is usually represented by x and the dependent variable by y.

Negative and Positive Slopes

Negative and positive slopes you can 100% easily tell by looking at the equation. But by looking at the graph it is not as simple but easy to learn. As you can see the very left graph is going up to the right. While the second left is going up to the left. Here are easy acronyms. If a graph at all goings up and right it is positive.

(UP RIGHT)UR=Positive

(UP LEFT)UL=Negative

(Down Right)=Negative

(Down Left)=Negative

Undefined Slopes Only X and Only Y

Undefined slopes are where an equations has only x but no y. For example x=-3. The x=-3 is the graph on the left and this is a relation. It cannot function as a graph with data because it would display different data at the exact same time. y=3 on the right is a function. It works perfectly fine as if a piece of data never changed but time moved on.

Parallel and Perpendicular Slopes

Rules for Perpendicular Slopes.

In Perpendicular the slope signs and numbers need to flip. If the slope is 3/4x it needs to turn into -4/3x.

Parallel Slopes are nice and simple. They don’t change at all. They stay the exact same except the y intercept may vary. If it is 3/5x+10 and 3/5x+5 it is the exact same they just start at different points. Both are positive, both go into the same directions. Just a duplication moved higher or lower.

In week 11 of math 10 we have learned x and y intercepts of equations with tables.

Example. 4x-2y+16=0. We have done something similar with this as we did last week. We need to get that “16” on the other side of the equal sign. We have learned to do that. In order to do so we have to do the inverse of addition to move it on both sides. We do this. 4x-2y+16-16=0-16. Then we get 4x-2y=-16. We have successfully moved the 16 by doing the opposite.

The next step would be being able to find the intercepts of either X or Y. To find the x-intercept here you would take the number next to the X and in this case it is 4. Next you divide it on the number on the other side of the equal sign. That being -16. You get 4/-16=-4. X-intercept=-4. For y it is the same. -2 is next to the y so we take -2 and divide it by -16. -2/-16=8.

We learned how to solve simple questions with Y and X intercept relations.

Example y=x-5. The y intercept of this question would be -5. Because we would need divide the quantity of Y’s by the end number. With here ONE y divide -5 is basically 1/-5 is just -5.

This would be the same with 2y=x-60. I take the 2 y’s and divide it by -60 making it 2/-60=30. The y-intercept would be -30.

There are some equations that look different but are the same. 2y+3x-12=0. You would need to take the 12 and add 12 to both sides of the equation to cancel it out on one side in order to move it. Then it would look like this. 2y+3x=12. Finding intercepts are the same. For y as we know we take the 2 from the 2y and divide it by 12. 2/12=6 which is the y-intercept. Now X is the same. take the 3 and divide it by 12. 3/12=4 and x-intercept is now 4.

Y=mX+b. “mX” being the slope and b is y-intercept. With run and rise.