8th Grade Math SC READY (SCCCR) /Expressions, Equations, and Inequalities
[8.EEI.8cd] Solve systems2 - Skill/Concept
Students in 8th grade Math in South Carolina are expected to demonstrate the skill and concept of solving systems of equations using different methods. This includes:
8.EEI.8c: Solve systems of two linear equations in two variables algebraically, graphically, and using matrix methods, and give examples showing how to use each method.
To solve a system of equations, you need to find the values of the variables that make both equations true at the same time. There are different methods to solve systems of equations, which include:
1. Graphing: You can graph both equations on the same coordinate plane and find where they intersect, which is the solution of the system.
2. Substitution: You can solve one of the equations for one variable in terms of the other variable and substitute this expression into the other equation. This will reduce the system to one equation with one variable, which you can solve. Then you can back-substitute this value into one of the original equations to find the value of the other variable.
3. Elimination: You can add or subtract the two equations to eliminate one variable and create an equation with only one variable. You can then solve this equation and use back-substitution to find the value of the other variable.
4. Matrix methods: You can write the system as an augmented matrix and use matrix operations to reduce the matrix to row-echelon form or reduced row-echelon form. The resulting matrix will give you the solution of the system.
Each method has its advantages and disadvantages, and some methods may be more appropriate for certain systems than others. It is important to be familiar with all methods and know when to use each one.
Examples of situations where you may need to use systems of equations include finding the intersection point of two lines, finding the optimal solution to a system of linear inequalities, or modeling real-world situations such as the amount of ingredients needed for a recipe.
To solve systems of equations with two variables, follow these steps:
1. Write down the two equations in the system.
For example, let's say we have the system:
Equation 1: 2x + 3y = 7
Equation 2: 4x - y = 2
2. Choose one of the equations and isolate one variable.
Let's take Equation 2 and isolate y:
4x - y = 2
-y = -4x + 2
y = 4x - 2
3. Substitute the expression for the variable found in Step 2 into the other equation.
Substitute y = 4x - 2 into Equation 1:
2x + 3(4x - 2) = 7
4. Simplify and solve the resulting equation.
Distribute 3 to the terms inside the parentheses:
2x + 12x - 6 = 7
Combine like terms: 14x - 6 = 7
Add 6 to both sides: 14x = 13
Divide both sides by 14: x = 13/14
5. Substitute the value of x back into either original equation to solve for y.
Let's use Equation 2:
4x - y = 2
Substitute x = 13/14:
4(13/14) - y = 2
Simplify: 52/14 - y = 2
Subtract 52/14 from both sides: -y = 2 - 52/14
Simplify: -y = 28/14 - 52/14
Combine like terms: -y = -24/14
Divide both sides by -1: y = 24/14
So the solution to the system of equations is x = 13/14 and y = 24/14 (which can be simplified to x ≈ 0.93 and y ≈ 1.71).