)    
        return " + " + printable(a) + x;
    if (a_trimmed < 0)    
        return " - " + -printable(a) + x;
}

function printable_eq(a1, x1, a2, x2, b) {
    if (Number(a1.toFixed(2)) == 0.0 && Number(a2.toFixed(2)) == 0.0)
        return "Not a valid equation; both coefficients are 0."
    if (Number(a1.toFixed(2)) == 0.0)
        return coefficient1(a2, x2) + " = " + printable(b);
    return coefficient1(a1, x1) + coefficient2(a2, x2) + " = " + printable(b);
}

function update_matrix(name) {
    i