algebra25 min

Solving Linear Equations

Isolate the variable to solve equations of the form ax + b = c using inverse operations

0/9Not Started

Why This Matters

A linear equation is the simplest kind of equation you can solve: it has one variable raised to the first power and its graph is a straight line. When you write 3x + 5 = 20, you are asking "what number, when tripled and increased by 5, gives 20?" Solving means isolating x on one side using inverse operations -- subtracting undoes adding, dividing undoes multiplying.

The concept of slope measures how steeply a line rises or falls. A line with slope 2 rises 2 units for every 1 unit you move right. The y-intercept is where the line crosses the y-axis, the starting value when x is zero. Together, slope and y-intercept define the equation y = mx + b, the most important formula in introductory algebra.

Linear equations appear everywhere in programming. Setting a variable equal to a computed value, solving for an unknown in a formula, and reasoning about rates of change all rely on the same skill: systematically undoing operations to find the value of the unknown.

Define Terms

Visual Model

Equation ax + b = cOne unknown, degree 1
Subtract bMove constant to right
ax = c - bVariable term isolated
Divide by aRemove coefficient
x = (c - b) / aSolution found
Check: plug back inVerify a*x + b = c

The full process at a glance. Click Start tour to walk through each step.

Solving a linear equation by applying inverse operations step by step to isolate the variable.

Code Example

Code
// Solve ax + b = c for x
function solveLinear(a, b, c) {
  if (a === 0) {
    return b === c ? "infinite solutions" : "no solution";
  }
  return (c - b) / a;
}

// 3x + 5 = 20  =>  x = 5
console.log("3x + 5 = 20:", solveLinear(3, 5, 20)); // 5

// -2x + 10 = 4  =>  x = 3
console.log("-2x + 10 = 4:", solveLinear(-2, 10, 4)); // 3

// 0.5x - 3 = 7  =>  x = 20
console.log("0.5x - 3 = 7:", solveLinear(0.5, -3, 7)); // 20

// Slope and y-intercept from y = mx + b
function lineInfo(m, b) {
  console.log(`Slope: ${m}`);
  console.log(`Y-intercept: ${b}`);
  console.log(`At x=0: y=${m * 0 + b}`);
  console.log(`At x=1: y=${m * 1 + b}`);
  console.log(`At x=5: y=${m * 5 + b}`);
}

console.log("\nLine y = 2x + 3:");
lineInfo(2, 3);

// Find x-intercept (where y = 0)
function xIntercept(m, b) {
  return -b / m;
}
console.log("\nX-intercept of y = 2x + 3:", xIntercept(2, 3)); // -1.5

Interactive Experiment

Try these exercises:

  • Solve 5x + 10 = 35 by hand. Then verify using the solveLinear function.
  • What happens when a = 0 and b = c? What about when a = 0 and b is not equal to c? Why?
  • Create a line with slope 3 and y-intercept -2. Compute y for x = -3, 0, 1, 4, 10.
  • Find both the x-intercept and y-intercept of the line y = -4x + 8. What shape do they form?
  • Solve 2(x + 3) = 14. First distribute, then solve. Verify by substitution.

Quick Quiz

Coding Challenge

Linear Equation Solver

Write a function called `solveForX` that takes three numbers a, b, and c, and returns the solution to ax + b = c. If a is 0 and b equals c, return the string 'infinite'. If a is 0 and b does not equal c, return the string 'none'. Otherwise return the numeric solution.

Loading editor...

Real-World Usage

Linear equations underpin countless practical computations:

  • Unit conversion: Converting Celsius to Fahrenheit uses F = 1.8C + 32, a linear equation you can solve in either direction.
  • Business break-even analysis: Revenue = Cost becomes price * units = fixedCost + variableCost * units, a linear equation solved for units.
  • Physics motion: Distance = rate * time (d = rt) is a linear equation. Solving for any one variable when you know the other two.
  • Proportional scaling: Resizing images, adjusting recipes, or computing tax all involve linear relationships.
  • Machine learning: Linear regression fits y = mx + b to data points -- the simplest predictive model.

Connections