Linear Equations
Understanding Linear Equations is crucial at the secondary 1 because they serve as a cornerstone for more advanced mathematical concepts. Linear Equations introduce students to fundamental algebraic concepts such as variables, coefficients, and constants. Mastering these basics is essential for progressing to more complex algebraic topics.
Linear Equation
A linear equation is a math statement where the variables (like x or y) are not raised to any power other than 1. In simpler terms, no squared or cubed variables.
Each linear equation has an equals sign, showing that both sides are exactly equal.
When graphed, linear equations make straight lines. They're often written in the form ax+by+c=0, or more commonly, y=ax+b.
For example:
- 3x−2y+6=0
- y=4x+3
- 5y−9x=6
- 5x=15
Solving Linear Equation
In this chapter, we will be discussing the below-mentioned topics in detail:
- Solving a Linear Equation by Balancing the Equation.
- Solving a Linear Equation with Fractional Coefficients.
- Solving Simple Fractional Equations that can be Reduced to Linear Equations.
1. Solving a Linear Equation by balancing the equation.
Case 1: x+a=c,where a and c are constants.
Let’s understand this with the help of some examples:
Question 1:
Solve: x+5=8.
Solution:
x+5=8x+5–5=8–5x=3
Case 2: ax+b=c,where a,b and c are constants.
Let’s understand this with the help of some examples:
Question 2:
Solve: 3x−1=5.
Solution:
3x−1=53x−1+1=5+13x=6
Dividing both the sides by 3
3x÷3=6÷3x=2
Hence, x=2.
Case 3: ax+c=bx+d,where a,b,c and d are constants.
Let’s understand this with the help of some examples:
Question 3:
Solve: 4x+1=2x−7.
Solution:
4x+1=2x−74x+1–1=2x–7–14x=2x–84x–2x=−82x=−8x=−4
Hence, x=−4.
Case 4: a(bx+c)=px+q,where a,b,c,p and q are constants.
Let’s understand this with the help of some examples:
Question 4:
Solve: 3(x+2)=x+14.
Solution:
3(x+2)=x+14
Expanding the equation,
3x+6=x+143x–x=14–62x=8x=4
Hence, x=4.
2. Solving a linear equation with fractional coefficients.
Case 5: xa=b,where a and b are constants.
Let’s understand this with the help of some examples:
Question 5:
Solve: x3=–5.
Solution:
x3=–5
Multiplying both sides by 3
x3(3)=(–5)(3)x=–15
Case 6: xa+b=c,where a,b and c are constants.
Let’s understand this with the help of some examples:
Question 6:
Solve: x4–2=3.
Solution:
x4–2=3x4–2+2=3+2x4=5x=5×4=20
Hence, x=20.
Case 7: abx+c=d,where a,b,c and d are constants.
Let’s understand this with the help of some examples:
Question 7:
Solve: 23x+1=7.
Solution:
23x+1=723x+1–1=7–123x=62x=18x=9
Case 8: abx+c=pqx+r,where a,b,c,p,q and r are constants.
Let’s understand this with the help of some examples:
Question 8:
Solve: 34x–3=x5+8
Solution:
34x–3=x5+834x–3+3=x5+8+334x=x5+1134x–15x=111120x=1111x=11×2011x=220x=20
3. Solving Simple Fractional Equations that can be reduced to linear equations.
Case 9: ax+bc=d,where a,b,c, and d are constants.
Let’s understand this with the help of some examples:
Question 9:
Solve: 2x+35=7.
Solution:
2x+35=72x+3=7(5)2x+3=352x+3–3=35–32x=32x=16
Hence, x=16.
Case 10: ax+bc=px+q,where a,b,c,p and q are constants.
Let’s understand this with the help of some examples:
Question 10:
Solve: 4x+13=2x–3.
Solution:
4x+13=2x–34x+1=3(2x–3)4x+1=6x–94x–6x=–9–1–2x=–10x=5
Hence, x=5.
Case 11: ax+bc=px+qr,where a,b,c,p,q and r are constants.
Let’s understand this with the help of some examples:
Question 11:
Solve: 2x–35=x+14.
Solution:
2x–35=x+144(2x–3)=5(x+1)8x–12=5x+58x=5x+173x=17x=173=523
Hence, x=523.
Case 12: ax+bc=pq,where a,b,c,p and q are constants.
Let’s understand this with the help of some examples:
Question 12:
Solve: x+12x−3=−15.
Solution:
x+12x−3=−155(x+1)=(–1)(2x–3)5x+5=–2x+35x=–2x–27x=–2x=−27
Case 13: ax+bc+px+qr=d,where a,b,c,d,p,q and r are constants.
Let’s understand this with the help of some examples:
Question 13:
Solve: x+12−x−13=1.
Solution:
x+12−x−13=13(x+1)−2(x−1)6=13x+9−2x+26=1x+116=1x+11=6x=–5