Linear Equations and Graphing
A LINEAR EQUATION makes a straight line when graphed. These equations describe thousands of real-world relationships — your phone bill, a car's gas usage, savings over time. Mastering them in 8th grade sets you up for Algebra 1 and beyond.
Slope-Intercept Form: y = mx + b
The most important equation in algebra:\n\n**y = mx + b**\n\n- **m** = SLOPE (how steep the line is)\n- **b** = Y-INTERCEPT (where the line crosses the y-axis)\n- **x** = any input value\n- **y** = the output\n\nEvery straight line in math has this form.
Understanding Slope
**Slope = rise / run** — how much y changes per step of x.\n\n- Slope = 2 means "up 2 for every 1 right"\n- Slope = 1/3 means "up 1 for every 3 right"\n- Slope = -1 means "DOWN 1 for every 1 right" (line goes down!)\n- Slope = 0 means the line is HORIZONTAL (flat)\n\nPositive slope = up. Negative slope = down. Zero = flat.
The Y-Intercept
The Y-INTERCEPT is where the line hits the y-axis — the value of y when x = 0.\n\nIn y = 2x + 5, the y-intercept is **5**. The line crosses the y-axis at (0, 5).\n\nIn y = 3x - 4, the y-intercept is **-4**. The line crosses at (0, -4).\n\nEasy trick: the constant term (not multiplied by x) IS the y-intercept.
In y = 3x - 7, what is the SLOPE?
Graphing y = mx + b
Step-by-step to graph y = 2x + 1:\n\n1. **Plot the y-intercept**: (0, 1). Put a dot on y-axis at 1.\n2. **Use slope to find next point**: Slope is 2 = 2/1. Go UP 2, RIGHT 1 from (0,1). Arrive at (1, 3). Put a dot.\n3. **Draw the line**: Connect the dots and extend.\n\nThat is it. Every linear equation graphs in 3 steps.
Real-Life Linear Equations
- **Phone plan**: Cost = $30 (base) + $0.10 × minutes → y = 0.1x + 30\n- **Taxi**: Fare = $3 (start) + $2 × miles → y = 2x + 3\n- **Savings**: Balance = $100 (start) + $25 × weeks → y = 25x + 100\n- **Speed**: Distance = 60 × time → y = 60x (b is 0 here)\n\nMany rates + starting amounts → linear equation.
If y = 5x - 2, where does the line cross the Y-AXIS?
Graph 5 Lines
On graph paper, graph these in different colors:\n\n1. y = x\n2. y = 2x\n3. y = (1/2)x\n4. y = -x\n5. y = 3x + 2\n\nLabel each. Compare:\n- Which is steepest?\n- Which goes DOWN?\n- Which shifted UP?\n6. Patterns will emerge.
Real-Life Equation
Think of a real situation that follows y = mx + b:\n\n1. Name the variables (what is x? what is y?)\n2. Pick a slope and y-intercept that makes sense\n3. Write the equation\n4. Solve for a specific case (example: if x = 10, what is y?)\n5. Graph it\n6. Algebra meets REAL LIFE — that is the power of linear equations.
A line has slope = 0. What does it look like?
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