Linear equations
Linear Equation
A linear equation is any equation that may be put in the following form:
ax+b=cx+d
Where x is a variable and a,b,c,d are real numbers.
The following equation is a linear equation:
2x-3=5x+7
Solving linear equations with simple operations
To solve a linear equation, isolate the variable using simple operations.
Consider the following linear equation:
2x-3=5
To isolate the variable x, begin by adding 3 to both sides of the equation:
2x-3+3=5+3
2x=8
Divide both sides of the equation by 2 :
\dfrac{2x}{2}=\dfrac{8}{2}
x=4
We have now isolated the variable x and found that x=4 is the solution to the linear equation.
Solving linear equations with graphs
Consider a linear equation:
ax+b=c
Where a,b,c are real numbers.
The linear equation can be solved by finding the x -coordinate of intersection of the lines y=ax+b and y=c.
Consider the following linear equation:
2x-3=5
Solve the equation graphically by finding the x -value of where the graph of y=2x-3 intersects the graph of y=5.
The point of intersection between the two lines is at the point \left(4{,}5\right). The x -value of the intersection is 4. Therefore, the solution to the linear equation is x=4.
Solving linear equations with variables on both sides
Solving a linear equation with variables on both sides of the equation requires isolating the variable.
Consider the following linear equation:
3x+4=x+8
To solve the linear equation, isolate the variable x. To do so, begin by subtracting x from both sides of the equation.
3x+4-x=x+8-x\\2x+4=8
Now solve the linear equation as before:
2x+4-4=8-4\\2x=4\\\dfrac{2x}{2}=\dfrac{4}{2}\\x=2
A linear equation with a variable on both sides of the equation can be solved graphically.
Linear equations with variables on both sides of the equation can be solved graphically. Consider the following linear equation:
3x+4=x+8
To solve the equation, graph y=3x+4 and y=x+8. The x -value of the point of intersection is the solution to the linear equation.
The point of intersection is at \left(2{,}10\right) : the x -value of the point of intersection is 2. Therefore, the solution to the linear equation is x=2.
Solving linear equation with algebra tiles
Algebra Tile
An algebra tile is a colored rectangle.
- A yellow rectangle represents a positive number.
- A red square represents a negative number.
- A green rectangle represents a variable.
Algebra tiles can be used to solve linear equations.
Consider the following linear equation:
2x-5=3
The linear equation above can be represented using algebra tiles.
Positive numbers cancel negative numbers. Therefore, red and yellow tiles cancel each other out.
To solve for the variable x in the linear equation 2x-5=3, add five yellow tiles to each side of the equation in order to isolate the green tiles.
The yellow square and the red squares cancel on the left hand side of the equation.
To solve for the unknown variable, which is the green tiles, rearrange the yellow tiles into two rows of four.
Each green tile has been aligned with four yellow squares.
This means that the solution to the linear equation 2x-3=5 is x=4.